There's some super basic research being done on space elevators, but, last time I looked, it was still mostly material studies and a couple of super small experiments.

From what I've seen there are still a number of material and architectural breakthroughs needed to make a space elevator viable, and my gut feeling is that seeing one within the next 50 years is probably overly optimistic.

I think we're going to be stuck with chemical propulsion for the foreseeable future.