Since following technology is a hobby for me, I follow the technology development that's fun to think about succeeding, even if the odds are low.
One of my favorites is the EM Drive. It's a proposed thruster for spacecraft which claims to produce thrust without reaction mass. It turns pure electricity into linear thrust, without having to throw anything out the back of the ship. There are similar research projects around the world looking into what seems to be the same phenomena, including Paul March's work on a Mach-Lorentz thruster, the Cannae drive invented by Guido Fetta, and replication work of the EM Drive being done by some researchers in China.
If the EM Drive or something like it ever works, it would initially not help us reach orbit. The thrust is too low to escape Earth's gravity. You'd still need chemical rockets to reach space. Once you're in space though, things change. A small but steady thrust can add up to very high velocity, helping making trips to Mars or further out much faster. Instead of reaching the outer planets in years, it would only take a few months, much like a ocean voyage during the age of tall ships. Further, since you don't need to bring reaction mass with you (and most of a space craft's mass is currently devoted reaction mass and/or oxidizer) it opens up the mass budget of any space mission to add more crew or science payloads.
Here's a current summary of where EM Drive research is sitting. (Be sure to reach the comments. Next Big Future is the rare website where the comments are almost always value-ad) The very short summary is that very small thrusts have been measured at NASA facilities, and now they're trying to scale it up a bit. If they can get it to scale up even a little it will be immediately useful on satellites and space stations for help maintaining their orbit. And then with a little more scaling, the whole solar system becomes available for exploration and settlement.
(And if it can continue to scale, further out you might eventually get that X-34 Landspeeder).