How many Kingdoms are there? There were 5 when I was studying in the 80s, now there seem to be anything up to 8, and a new higher division called Domains? What is current thinking on this?

Ah... this all depends a bit on where you are and/or which book you choose to give you your definitive answer. Domains were proposed in 1990 by Woese and coworkers, and have Kingdoms within them. Many consider them useful and have adopted Woese's suggestion, but different people have different ideas of course! Most biologists really consider these high level phyla to as much an organisational "filing" system for life as much as anything else. Accepting that view it doesn't seem surprising that some people prefer one organising system whicle others disagree. An important point is that,  at least as previously used, Kingdoms are not monophyletic and therefore sit rather uncomfortably with a lot of modern taxonomy anyway  -to the point where some people think its easier not to bother teaching this stuff!

Rather than give you a full rundown here I'll just refer you to the wiki page which looks to do a good job of summarising.