What type of Begonia? There are lots of varieties. Some have tubers that you store, others are really annuals and others are perennial and you need to bring inside. Is there a label on it? If you did not pay much for them then my guess is you have annual ones.

If they are tubers allow them to dry off. Remove them from the compost and cut off the stem. Store the tubers in dry sand, peat compost etc.somewhere cool where mice and frost can't get at them.

If they were bedding begonias then as Kildermorie says they are treated as annuals in this country. You can keep them alive as house plants but its not practical to keep them all!