Currently functional and proven technology is limited to basically no interstellar travel at all. To reach one of our stellar neighbors (like Proxima Centauri), one of the fastest space probes we have now, New Horizons, would take 54000 years.
There are multiple proposed methods of sending spacecraft interstellar distances (in shorter time spans) such as:
Beaming power (use a laser to power/push the craft)
High energy fuel (nuclear/antimatter?)
Electric propulsion (Ion drives or other electric thrusters)
Picking up fuel along the way (Interstellar 'ramjet')
Cannon / projectile (Providing speed by launching in a railgun or similar)
Theoretical / Sci-fi (Warp drives, wormholes, etc.)
Of these methods, beaming power is probably the closest to reality right now. For example there is a proposal for sending small coin-sized spacecraft to other stars using a 'sail' and a powerful array of lasers to speed it up. This is what the "Breakthrough Starshot" proposal is and it could accelerate the tiny probes up to about 20% of light speed allowing the probes to reach the star in about 20 years. This system would work however there's still some engineering challenges and funding challenges (nothing impossible though). Such a system would be only a one way trip without stopping at the end--the probes are too small to do anything besides collect basic sensor information and since they have no propulsion, most of them would fly by and continue into deep space. The problem with the Breakthrough Starshot is that it's limited to extremely small spacecraft currently however future developments in laser and power generation technology may increase the size and speed of the space probes we can send. According to the organization, if funded, they could launch the first probes in 2036.
Next on the list of closest-to-reality, the high energy fuel category has certain proposals would be build-able today, albeit at enormous cost, risk, and engineering challenge such as the Daedalus Interstellar Craft. This spacecraft was proposed back in the 1970's. Using nuclear propulsion it would be able to accelerate up to ~12% of light speed resulting in a trip duration of an estimated 50 years to our near neighbors.
Progress is also being made on electric propulsion which is making it more and more practical for in-system transportation however the energy requirements it has and low thrust make it sub-optimal for fast interstellar flight.
Other methods such as interstellar ramjets or warp drives are currently only fun in the lab and while mathematically possible aren't currently translatable to reality.
Breakthrough Starshot's goals of first launch by 2036 and thus first arrival around 2056 are reasonable, although only if they get the funding. As for larger (or manned) interstellar missions, don't expect any within the next 50-100 years unless some breakthrough technology is developed (like human hibernation/stasis systems or a practical new propulsion method).