If you're in the US, Fox Soccer 2Go streams the Champions League, the Europa League, the English FA Cup, and (new this year) the full Bundesliga 2015-16 season, among other competitions. The quality isn't great, but it's reliable and pretty cheap ($99/year) compared to a US cable subscription that would carry the equivalent competitions. They also keep full match replays online for a few weeks after the live broadcast, for some competitions, anyway. You can watch on a PC/Mac in a browser with Flash (ugh), or use their Android or iOS app. I haven't used the Android app, but the iOS app, despite its awful UI, works fine and supports AirPlay to an Apple TV.

Also in the US, fubo.tv carries some matches in a few other top European leagues. It's a bit of a strange service, but it's cheap and worked well enough for me the few times I used it last season. The quality is slightly better than Fox Soccer 2Go, but still nowhere near broadcast HD. They appear to have a deal with BeIN Sports such that, if a match is being shown on TV on BeIN Sports USA, you can stream it online with just a subscription to fubo.tv (no TV service required). I can't figure out BeIN Sports USA; it seems like they show some Serie A matches, some Ligue 1 matches, and some La Liga matches, but not all. fubo.tv used to carry Bundesliga matches, as well, but as that's moving to Fox Soccer 2Go this season, I don't think fubo.tv will carry those anymore. In any case, I'm waiting until August to decide whether or not to renew my fubo.tv subscription, based on what they're offering for the upcoming European football season.

The Barclays Premier League is NBCSN-only in the US and requires a cable subscription -- for now, at least.

Links:

http://www.foxsoccer2go.com/competitions http://www.fubo.tv