DOWNLOADING music and video files, without the permission of the copyright owner (or without paying to obtain such permission) is a civil offence, for which you can be sued (but not fined or imprisoned - unless you do so on a very large scale or by way of business).
However the way that P2P systems work is that files on your own computer are made available for others to access. That's UPLOADING or, as the law defines it 'DISTRIBUTION'. As well as being a civil offence (for which you can be sued), it's also a criminal offence (which can lead to fines or imprisonment).
Most of the prominent legal cases have occurred in the USA:
http://news.bbc.co.uk/1/hi/8478305.stm
http://news.bbc.co.uk/1/hi/7029229.stm
http://news.bbc.co.uk/1/hi/8177285.stm
In the UK, most users normally agree to make a nominal payment (often in the region of £1000 to £2000) rather than face court action.
Chris