Beam Me Up
The Movie: Like all Star Trek incarnations before it, Into Darkness features a whole ton of beaming up, down, sideways, and now, "transwarp."
Real Life: Teleportation is another science fiction staple that we must suspend our disbelief over, because it adds so much to a fictional universe. Star Trek simply wouldn't be Star Trek without it. But how realistic is it? The good news is teleportation of photons has already been achieved (even though photons are a very weird type of particle, being massless and all).
The bad news is that these experiments do not "jump" the photon across space; they make a copy of the photon in a new place, causing the old one to be destroyed. So if Captain Kirk went through this process, a copy of him would be beamed to an alien world, and the universe would freak out about the original and annihilate it. This is problematic because Kirk is a lot more complicated than a photon. Our bodies contain about a trillion trillion atoms, each of which would have to be replicated perfectly in another place. If the copy was unsuccessful, tough luck, because the original is dunzo. It's been proven over and over that you should never say never when it comes to science, but there are definitely some major problems to overcome in this case. Sorry, nerds.