Continuing the trend of Egann is the only person to start threads in S&H, we've directly observed gravity waves!
There are quite a few sources online, but I like the DNews best. They actually transpose the signal detected into audio. Listening to two black holes totaling sixty times the size of our sun merging and making a "bloop" sound from over a billion lightyears away is AWESOME.
Applications? Study stellar events and the big bang, of course, but I want to throw one other thing out there I haven't been hearing people talk about.
Practically every theoretical FTL drive we can think of works by manipulating space-time in some way, which means it will produce gravity waves. Not big ones like the black holes merging did, of course, but it's still something we can directly detect. Heck, if we get a good enough signal...we might be able to figure out what the drive is doing to space-time and reverse engineer it.