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According to a recent news article a group of USC students are attempting to launch a rocket "in to space" with a planned height of 62 miles. Making them "the first group of students to successfully launch a rocket into space,"
The ISS being at about the 250 mile height, and weather balloons reaching heights of 25 miles or more, space would seem to begin someplace between those two heights. Where does "space" begin?
We are "in space", in fact everything that exists and has a physical presence is. But what we usually mean by it is to describe "outer space" conditions of near (or hard) vacuum, where atmospheric pressure is already low enough to affect matter differently than under true atmospheric conditions, for example at atmospheric pressure below triple point of water. But where "outer space" begins, as you might expect by now, is a matter of opinion and there isn't any clear line to be drawn. For example, Wikipedia on Outer space has this to say:
There is no firm boundary where space begins. However the Kármán line, at an altitude of 100 km (62 mi) above sea level, is conventionally used as the start of outer space in space treaties and for aerospace records keeping.
Kármán line, named after the Hungarian-American engineer and physicist Theodore von Kármán, is the theoretical altitude at which the Earth's atmosphere becomes too thin for aeronautical purposes. But this is merely one arbitrarily set boundary, and where "outer space" begins could as well be defined differently, for example by the mentioned atmospheric pressure where water can sublimate directly from ice to gas phase without the transition to a liquid first. Again quoting Wikipedia, this time on the Triple points of water:
The gas–liquid–solid triple point of water corresponds to the minimum pressure at which liquid water can exist. At pressures below the triple point (as in outer space), solid ice when heated at constant pressure is converted directly into water vapour in a process known as sublimation.
So there it is again, mentioned as where the "outer space" begins. And it could, of course, be defined by other means as well. For example by what arbitrary atmospheric boundaries we set, at the edge of any altitude that we name atmospheric layers with a different term, designating different atmospheric conditions, maybe at the edge of Stratosphere and Mesosphere at 50 km? Yet again quoting Wikipedia on Uncertainties of Mesosphere:
The mesosphere lies above the maximum altitude for aircraft and below the minimum altitude for orbital spacecraft. It has only been accessed through the use of sounding rockets. As a result, it is the most poorly understood part of the atmosphere. The presence of red sprites and blue jets (electrical discharges or lightning within the lower mesosphere), noctilucent clouds and density shears within the poorly understood layer are of current scientific interest.
So it again mentions the Kármán line, but gives us an additional clue that "outer space" might as well be defined by where atmospheric weather phenomena stops. This could go on forever really, so I'll stop with these three examples.
We can draw a few conclusions though:
Concurrent with TildalWave's reply I too say 'Karman line'(100 kilometres (62 mi) above the Earth's sea level). Apart from the fact the K-line is legally so recognized an alternative definition of 'in space' is covered in the same Wikipedia article
... any vehicle at this altitude would have to travel faster than orbital velocity in order to derive sufficient aerodynamic lift from the atmosphere to support itself, neglecting centrifugal force). ...
The beginning of outer space and spaceflight is something hard to classify because the boundary between an atmosphere and the vacuum of space is very fluid. A space border is to be defined according to what one considers air and space / what one considers important in these matters. The space border may be an air pressure border rather than an altitude border, and should apply to all celestial bodies.
One of the following altitudes and pressures might be set as a border between an atmosphere and outer space / between air- and spaceflight (some of them are based on my personal spaceflight experience in Orbiter2016):
60,000 ft (18.3 km) The Armstrong limit above which the outside air pressure is so low that you need a pressurized suit (like a spacesuit). Water would boild at the temperature of your body. So your body considers the space above the Armstrong line to be vacuum and you can't survive without a pressurized suit or cabin anymore. 90% of the Earth's atmosphere's mass is below you. FAA airspace legislation ends at the Armstrong limit. The sky gets very dark already above 60,000 ft and you'd see the brightest stars and planets at noon. The Armstrong limit marks the begin of nearspace, a transition area between airspace and outer space. If you consider it the space border already, the following celestial bodies would count as bodies with atmospheres: Venus, Earth, Titan, and the four gas giants.
115,000 ft (35 km) The triple point of water. Above that altitude water can't exist in liquid state outside anymore. Water ice would sublime (evaporate), not melt. The triple point of water is at a pressure of about 611.7 pa (0.088 psi). At this altitude there's also the upper boundary of the ozone layer above which there's little block of UV radiation. Above that altitude the sky is completely black and won't get any more blacker. You'd see all bright enough stars and planets at noon (such as the Orion in summer). Jet aircraft can't fly leveled anymore and the altitude record for a jet plane is a MiG-25M about 8,000 ft (2.5 km) above that altitude. If you consider the pressure of the triple point of water to be the space boundary, you are to add Mars to the list of bodies with considerable atmospheres.
32 mi (51.5 km) The stratopause (stratosphere-mesosphere border). Temperature stops increasing and starts decreasing with altitude. Above 32 mi, air pressure drops below 0.01 psi. If you consider this or a lower altitude the space border, note that Yuri Gagarin wasn't the first man in space for you. The first man in space would be American pilot Joseph Walker who reached slightly more than 32 mi in the X-15 on 30 March 1961, a few days before Gagarin's spaceflight.
200,000 ft (61 km) As I conclude from flying in Orbiter2016, above about that altitude, pressure drops below 0.003 psi. There it is so low that you can't hear anymore, there is no sound and one is essentially deaf above that altitude. Outside only, since sound would still travel through your spacecraft of course. Also above circa 200,000 ft the ionosphere begins. Balloon flight is no longer possible. The highest unmanned balloon reached an altitude of 173,900 ft (53 km) and the highest manned one (flown by Alan Eustace) reached about 136,000 ft (41.5 km). Above 200,000 ft you may become weightless in your spaceplane without having to push the yoke. See this answer for clarification.
71 km (230,000 ft) This is circa the lowest perigee I reached in Orbiter2016 and continued orbiting the Earth. The orbit didn't change much, it remained pretty stable.
50 mi (80.47 km) This is the space border as defined by NASA, the USAF and the FAA. It is the mesopause (mesosphere-thermosphere border): temperature stops decreasing and starts increasing again. Pressure falls below 1 Pa / 0.00015 psi above that altitude. It is defined as where you have to put more effort into rocket-powered flight rather than air buoyancy. Astrodynamics take over from aerodynamics around that altitude. If you consider the space border here, you must add Pluto, Eris and Triton to celestial bodies that have a considerable atmosphere.
83.6 km (51.9 mi) Theodore von Kármán calculated that at that altitude the atmosphere becomes too thin to support aeronautical flight.
53 mi (85.3 km) This is circa where in Orbiter2016 my spacecraft starts to glow when re-entering from orbit. I suppose the Space Shuttle started to glow around that altitude too. I regain rudder control around that altitude.
57 mi (91.5 km) The original Kármán line: a vehicle's speed to generate lift must be orbital speed. Aerodynamic lift is 2% while 98% of the vehicle's weight is carried by centrifugal force. While circular orbits are impossible at that altitude, an spacecraft in an elliptical orbit can attain a perigee at 230,000 ft and remain it quite stable.
100 km (62.14 mi) What is currently referred to as the Kármán line and set by the FAI as space border. It is just the nextmost double-0-value in metric units in order to make the "Kárman line" "more memorable", lacking any basis in physical properties.
65 mi (105 km) In Orbiter2016, my spacecraft's gravimeter begins to read a more considerable g-force around that altitude when re-entering from orbit (or when having an elliptical orbit with a perigee reaching that low).
118 km (73 mi) Quote from Wikipedia 1: "In 2009, scientists reported detailed measurements with a Supra-Thermal Ion Imager (an instrument that measures the direction and speed of ions), which allowed them to establish a boundary at 118 km (73 mi) above Earth. The boundary represents the midpoint of a gradual transition over tens of kilometers from the relatively gentle winds of the Earth's atmosphere to the more violent flows of charged particles in space, which can reach speeds well over 268 m/s (600 mph)."
120 km (75 mi) This is where in Orbiter2016 my spacecraft starts to experience significant atmospheric drag when re-entering from orbit. If you set the space border at this altitude or higher, you must include Io into the list of bodies with considerable atmospheres.
400,000 ft (122 km) NASA's re-entry altitude for the Space Shuttle, defined as the beginning of more significant atmospheric drag.
93 mi (150 km) Above that altitude, a stable circular orbit is possible.
450 mi (700 km) The thermopause / exobase (end of collisional atmosphere). Above circa that altitude the atmosphere becomes an exosphere which no longer behaves like gas. The molecules don't collide with each other and are dispersed away from Earth by solar wind, reaching escape velocity. If you consider this the space border, you must include Callisto into the group of bodies with considerable atmospheres. You would also have to classify only the following flights as spaceflights: Gemini 10, Gemini 11, Apollo 8 and Apollo 10-17. All other spaceflights wouldn't count as any.
10,000 km (6,214 mi) End of the exosphere. Above that altitude there's quite an absolute vacuum. If you consider this the space border, only Apollo 8 and Apollo 10-17 would you have to count as spaceflights.
35,786 kilometres (22,236 miles) Geostationary orbit. While this has nothing to do with air pressure/density and vacuum, some equatorial countries claimed legal right on the territory up to the Geostationary orbit altitude.
As for me, I consider 200,000 ft (61 km) the space border. The least plausible to me are the one at 100 km and the one at geostationary orbit, due to the reasons written above.
To contradict everyone else, the US Air Force used an altitude of 50 miles when awarding the Astronaut Badge to pilots of the X-15.
Outer space begins at the Kármán line (correctly stated by @TildalWave). This has been accepted by Fédération Aéronautique Internationale (FIA). Quoting Wikipedia:
The Kármán line, or Karman line, lies at an altitude of 100 kilometres (62 mi) above the Earth's sea level, and commonly represents the boundary between the Earth's atmosphere and outer space. This definition is accepted by the Fédération Aéronautique Internationale (FAI), which is an international standard setting and record-keeping body for aeronautics and astronautics.
But it is not where atmospheric weather phenomena ceases. Weather phenomena typically ceases in the thermosphere which reaches 8-15 km above the Earth (18 km above equator), but stratospheric weather phenomena does exist, e.g. nacreous or polar stratospheric clouds which can form as high as 25 km above the mean sea-level. Still, that's far below the Kármán line at 100 km above the Earth's surface.
Following wrong remarks about "the" Kármán line and a beautiful round number in metric units, I decided to add my two cents to this, or something.
First, concerning the Kármán line, let me tell you that the Earth doesn't have a Kármán line. It's the airplane in question that has its own Kármán line, this is where its stall speed reaches orbital velocity. Hence the Kármán line is different for each plane, but it's most often about 10 km below the alleged 100-km-value (hence around 90 km (56 mi)). Not that any plane would attempt to fly horizontally in the upper mesosphere or above by propulsion and aerodynamic lift only.
So back to your question now, how far from Earth is space, or better: At what altitude/distance does the atmosphere end?
Now, take a look at the following diagram:
Alternatively:
Atmospheric composition with altitude 1
Alternatively:
Atmospheric composition with altitude 2
As we can deduce from the diagrams, the change in atmospheric composition starts around 90 km (56 mi) altitude which is called the turbopause or homopause, and identical with the mesopause; and the change essentially ends around 880 km (550 mi) above which hydrogen and helium prevail.
Hence if you're strict you may state the Earth's atmosphere's definite end is at 550 miles, beyond which there is outer space. However, this would mean there were only ten human spaceflights in history: Gemini 11, Apollo 8 and Apollo 10-17. While this is an honor to America as it's the only nation that would ever have accomplished human spaceflight, textbooks and almost everyone consider Yuri Gagarin the first human in space, and the German V2 missile the first man-made object in space.
During its Peenemünde tests the highest altitude reached by the V2 was 189 km (117 mi). Yuri Gagarin's Vostok spacecraft reached an apogee of 327 km (203 mi). The previous altitude record for a human was 51.5 km (32 mi) reached by Joseph Walker in the X-15 a few days before Gagarin's flight. So if you'd like to stick to the mainstream, the atmosphere-space border should be above 32 miles and no higher than 117 miles. As concluded from the diagrams, the change in the atmosphere's composition begins at the turbo-/mesopause which is 84-92 km (52-57 mi) high, limiting a "mainstream border" even more.
But if you don't mind about mainstream teaching, you may consider the atmosphere-space boundary at any altitude from 84 km (52 mi) to 880 km (550 mi), or perhaps you'd like to define it by entirely different factors.
