There are 2 types of Begonia, Mike. Fibrous rooted (the ones with small flowers) and tuberous (the ones with big flowers).

I've tried keeping fibrous rooted begonias over winter and they don't do well, so best binned - in the compost heap.

Tuberous begonias, as you would expect, grow from a tuber that should be on the surface of the earth or compost. You can dig the plants up - say at the end of October - let the tuber dry off for a few days, then clean it up gently by brushing the earth off. I always let the stems fall off on their own. Then keep the tubers in a cool dry (frost free) place away from mice and check occasionally for vine weevil grubs that like to eat them in storage.

Next March you can start them growing again. Press them into the surface of some moist compost and bring them indoors, or into a greenhouse to get going.