Space elevators fascinate me. They've been a subject of debate since the late 1890s when Konstantin Tsiolkovsky proposed the first version of the idea: A tower so high that it reached geostationary orbit. (Later designs switched to a satellite station in space, with an extremely long tether connecting it to a base station on Earth.) The idea of a space elevator is that it would radically reduce the cost of getting stuff into space, because you'd simply climb the tower (or tether) rather than blast rockets to escape gravity..
Despite discussing space elevators for more than a century, we haven't managed to actually build one. Is it even possible? What would we build that super-long tether out of? What else is stopping this thing from being built?
Check out the transcript if you're in more of a reading mood.