As I recently discovered whilst trying to teach my housemate about torrenting — educationally, of course — downloading illegal stuff off the internet still isn't the easiest thing in the world. Bitport is a website that's trying to change all that: it will find, download, and stream your torrents to you, stupidly securely and easily, provided you pay for it.
Using Bitport is pretty simple. Once you've got an account (either the basic free one, with limited storage, or one of the per-month premium options), you log into a web console, from where the magic happens. You can link to a torrent file or magnet link, or upload a file yourself. The torrent downloads (you can do a bunch at a time, or queue downloads for later), and once the file is saved in your personal cloud storage account, you can download it, or more likely just stream the file online.
Torrenting the file online has a few major benefits for the user. The big one is speed: because the torrent is being downloaded by a server sitting in a data centre, things happen much faster: a years-old file with a couple seeders got pulled down at 10MB/s, for example. In the case of popular torrents, the download is instantaneous (i.e, if another Bitport user has downloaded the same file recently, it seems to appear straight away in your storage account, and you can stream instantly).
Privacy is another benefit: because files aren't stored on your personal computer, and connections are encrypted, it would be much harder for the movie studios to cut off infringement. And, if you're particularly paranoid, you can use a VPN or Tor to access the service.
One thing worth bearing in mind is that there's a similar website, which has been around for a while, called Put.io. It offers most of the same services — but notably, doesn't have a free tier, although you do get more storage for your money in the higher tiers. Annoyingly prices on BitPort are in Euros for us, but it'll cost you 4€-12€ (£3-£9). Those prices get you between 30GB and 250GB of storage.