X-Com, after UFO:TFFD has become more significant as the governments realized they cannot ignore the alien threat. X-COM took to the sky and space, leading the charge to conquer and colonize Mars. This was done through UFO:Interceptor, which I would play if it ran for longer than 5 minutes in Windows 7
Humans have come out extremely victorious, conquering Mars and setting up a very lucrative alloy, technology and elyrium trade. The new X-Com in space grew in importance and influence, and became Marsec - private security and arms trader. That's right, Marsec is X-Com 2.0. The current X-Com roster was kept essentially as an atrophied museum relic, in memory of what they have achieved. Essentially a cushy office job where you do nothing and scan for aliens.
The tech and especially elyrium trade has produced an enormous technological boom on Earth, producing all sorts of results. Mostly negative, to be honest. Psionics have become very wide spread as entertainment and similar with psiclone being a more... specialized product.
The Alien invasions started some time before the game starts, and X-Com had quite a kick in the pants where antiquated infrastructure had to suddenly deal with dimension gates and a real job to do. It failed spectacularly, and they flailed to find a new manager. That is, you.