The ship's crew is all killed by a careless mistake of Rimmer's, resulting in a radiation leak, and since Lister was in stasis he was unaffected. It took 3 million years for the radiation to die down, and Holly then woke Lister up. They are heading back to earth and running into all kinds of crazy adventures along the way. They are aboard Red Dwarf from the 1st to 5th series, and then they are chasing it in Starbug, a smaller ship, in the 6th and 7th. They get Red Dwarf back at the end of the 7th Series.
In the 7th Series, Rimmer goes off to become his alternate universe/dimension (whatever) self, Ace Rimmer, a more handsome, respected, cooler, more successful version of himself. The original Ace is long dead and has started a chain of "regular" Rimmers trained to take on his personality. Our Rimmer has gone to do just that. He does appear as normal Rimmer in the first two eps of the series, the second is the one where he leaves. He also appears in two more eps out of the 8, but in these he is only in flashbacks or virtual reality simulations.
Kochansky, Lister's love and ex girlfriend, has come onboard also in this series. However, this Kochansky is from another dimension, and wants to get back. She misses "her own" Lister very much which makes Lister change his old disgusting habits. Kryten is VERY jealous of Kochansky being aboard and the way it is affecting Lister, and is afraid they will get married and send him off the ship.
My favorite character is Rimmer, although they are all hilarious! Series 8 is currently in the making and I cannot wait to see it. I hope I get to! Rimmer is supposed to be back for all the eps. Also there is talk of a RD movie being made sometimes soonish.
Go to the Pile 'O Smeg website. Tons of links, and you can rate my site!
Here are a few pics of various parts Rimmer has played and also some group pics of the crew.