What are the laws on copying copyrighted material for personal use?

I know copyright laws can be complex and vary from region to region (I’m in the UK). I ask because when the cassette tape became popular the music industry was in uproar and had all sorts of adverts stating it was ‘killing the music industry’ I even have a t-shirt with a tape and cross-bones with the slogan It Is Killing It underneath But I believe the courts agreed that if you owned the LP you could record it to tape for your own personal use, in your walkman or car. Then we had the same with the writeable CD etc

I’m sure though that if you buy a DVD you should be able to copy it to any portable device capable of playing the media for your own use, legally? Am I correct or am I completely wrong?