There are 5 lagrangian points.
Are there any man-made satellites at any of those points? Is there a reason for the presence or absence of satellites at these points?
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2$\begingroup$ Wikipedia article you link to already has a list of missions to those points. $\endgroup$– RuslanCommented Nov 16, 2014 at 19:21
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$\begingroup$ Just to clarify one thing; Are you exclusively asking about the Sun-Earth libration points, as the image you include suggests, or any libration points? That's in particular relevant to the latter part of your question: Is there a reason for the presence or absence of satellites at these points? Please edit to clarify. Also, the answer to the question in the title is trivial, as already stated in the previous comment. Answers merely focusing on that also won't age all too well. Could you please rephrase to focus on the non-trivial parts of your question? Thanks! $\endgroup$– TildalWaveCommented Nov 16, 2014 at 23:29
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$\begingroup$ @TildalWave Sun earth primarily. Earth-moon is also nice. If there happens to be a man-made satelite at any other Lagrangian point regardless of the planet that would also be interesting. I wouldn't be the least bothered if someone shared there was a trinary star system that utilized Lagrangian points. Consider this a collaborative process as much as possible. I realize moderators don't always share my view but usually they just complain about this sort of productive and helpful assistance. $\endgroup$– DaleCommented Nov 16, 2014 at 23:59
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First of all you need to specify the bodies you are considering.
There are 5 Lagrangian points in the Sun-Earth system and there are other 5 Lagrangian points in the Earth-Moon system.
Basically you can find 5 Lagrangian points (which are equilibrium points int the synodic reference frame) for each system made of 3 bodies where $m_3 << m_1,m_2$.
In the Sun-Earth system, L1 is preferred for Sun observation missions, whereas L2 is good for deep-space observation since it is possible to have the Sun always behind the spacecraft (which could be a telescope for instance) therefore it is possible to observe the whole emisphere without having the Sun in the field of view.
L4 and L5 are stable, which means that once you get there you do not have to spend propellant to stay there, whereas the others are not stable therefore you need to adjust your orbit nearly every period not to be driven away.
Missions in L1: Explorer 3, Genesis, Wind, LISA (coming soon).
Missions in L2: Herschel&Planck, James Webb Space Telescope (coming soon hopefully)
In the Earth-Moon system L2 is used for telecommunication satellites.
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3$\begingroup$ The sun-earth L4 and L5 are definitely achievable with existing propulsion. They take much less than solar escape, which has been demonstrated with Pioneer and Voyager. We just haven't had a mission where they were chosen. The Wikipedia page lists some proposed missions to earth-moon L2, but none that were done. ARTEMIS went by, but didn't stay. $\endgroup$ Commented Nov 16, 2014 at 16:15
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$\begingroup$ Are you talking about travelling a distance equal to the L4-Earth distance or about actually orbiting around L4 (or L5)? Because of course we can travel that distance, but L4 and L5 are the highest energy equilibrium points therefore to get there you need much more energy than the one you need to get to L1 which is the location of the lowest energy equilibrium point $\endgroup$– RheiCommented Nov 16, 2014 at 16:52
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5$\begingroup$ It is still far less energy than solar escape, so we have plenty of propulsion to get to L4 or L5 if we want to. You can use lunar assist to make it about the same as launching something to the moon. The reason we haven't done it is lack of a compelling reason to do so. $\endgroup$ Commented Nov 16, 2014 at 16:56
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3$\begingroup$ @AnthonyX, Do they accumulate debris? Yes: the Trojan asteroids. Do they present a hazard? Unknown. $\endgroup$– MarkCommented Nov 17, 2014 at 1:34
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1$\begingroup$ -1 for the sentence L4 and L5 cannot be reached at the moment because the energy needed to go there is too high for the today propulsion technologies. That's nonsense. Remove that statement and I'll change my down-vote into an up-vote. $\endgroup$ Commented Nov 17, 2014 at 19:38