Obviously, it is illegal in most countries to download films, games, music etc without paying for them. However, there are a growing number of people who argue that streaming is not illegal as, they claim, that because streaming does not save the files after you have viewed them (despite there being caching) it does not count as downloading.
What does the law have to say on this?
Yes, it is legal in the European Union. https://en.wikipedia.org/wiki/Copyright_aspects_of_downloading_and_streaming
Ever since the Filmspeler case, the answer to this question has been ruled as "no". See also this writeup in the popular media.
Worth noting though that once the UK leaves the EU, it can make its own laws and this might no longer apply.
The other answer bases its legality on the fact that the CJEU has ruled it legal to view online content.
However, from the other perspective, it is still most certainly illegal to host illegal content. The laws in various EU member states deal with this.
Who says it is illegal to download copyrighted works without paying? See for example
http://www.blockmrecords.org/bach/
where you can download the complete organ works of Johann Sebastian Bach, performed by James Kibbie, legally, and for free. In this case only the recording falls under copyright anymore, but at
http://www.npr.org/2013/03/01/173275533/the-mix-the-austin-100
exactly 100 songs, about 6 1/2 hours of recordings, were freely and legally available for a considerable time.
The question isn't whether you have paid or not. The question is whether you have a valid license or not.
Just saying: The complete Bach organ works are still available. From the site: "This project is sponsored by the University of Michigan School of Music, Theatre & Dance with generous support from Dr. Barbara Furin Sloat in honor of J. Barry Sloat. Additional support has been provided by the Office of Vice-President for Research, the University of Michigan."
Or look at: https://www.bbc.co.uk/iplayer . I'm downloading movies there for free all the time.
