Manned interstellar space flight is not going to happen anytime soon. Definitely not within the next 50 years, most likely not within the next 100. There are too many basic issues that are hard to work around.
The biggest issue (apart from politics and funding) is propulsion. So far, nobody's figured out how to build an engine that doesn't require some form of reaction mass (propellant) to accelerate, meaning your total ΔV (change in velocity) is limited by how much reaction mass you can carry along, and the amount of reaction mass you need grows exponentially as ΔV increases (i.e., for N times more ΔV, you need eN times more reaction mass). Conventional chemical and ion engines are simply not up to the task of interstellar travel; either the amount of propellant necessary is many orders of magnitude larger than the spacecraft itself1, or you accept travel times of thousands to tens of thousands of years (which brings up issues with power, life support, spacecraft maintenance, etc.).
There are ideas for engines that use nuclear fission or fusion that provide better performance, but the best of them would still require on the order of a century or so to reach the nearest star. And again, you're limited by the amount of reaction mass you can carry.
The logical step up from that is to not carry your reaction mass with you; instead, you either gather reaction mass from the interstellar medium as you travel (like the Bussard ramjet), or you use a sail to exchange momentum with the solar wind or a powerful EM source, like a laser array. There's a project called Breakthrough Starshot that intends to use a massive bank of ground-based lasers to accelerate a gram-scale spacecraft with a large sail to something like 20% c.
The problem with ramjets is that there just isn't that much material to gather, and the drag from the scoops would probably cancel out whatever thrust you could generate. And the problem with sails in general (and Starshot in particular) is that you have no way to slow down once you reach your destination.
Basically, interstellar travel with reaction drives is not going to happen. We need to develop some kind of reactionless drive, one that doesn't require you to exhange momentum with some kind of reaction mass in order to accelerate.
There's the EmDrive, which allegedly provides thrust without reaction mass. There's a lot of skepticism associated with this, with good reason. At first blush, it seems to violate conservation of momentum. Also, people who've tried to replicate the results using sane test setups have not been able to reproduce the effect.
So, that pretty much leaves us with warp drives, which...aren't going to happen anytime soon, if ever. They're mathematically possible, but that doesn't mean that they will ever be physically possible, and even if they're physically possible, the energy requirements are such that we could probably never build one.
- To reach 1% c (3000000 m/s) using the ion engine on the Dawn spacecraft, you'd need on the order of 1088 kg of propellant for every kg of spacecraft mass. To reach that same velocity using an RL-10 chemical (hydrogen/oxygen) engine, you'd need on the order of 10295 kg of propellant for every kg of spacecraft mass.