I agree with Teresa,
Keep them frost free and dry. Last year, I bought a few begonia 'plugs' from my local garden centre. Cost me about 50p per tiny plant. They grew into lovely double bright scarlet begonias last summer and are now brightening up my patio. Twice the size this year!
Once all of the foliage has died back, carefully remove the corms (I think they are classed as corms??) from the compost and keep in a dry frost free place until spring.
I wonder how many people are given begonias for gifts and then throw them away, thinking that they are dead at the end of the season! :o