I used to wait until they no longer flowered, took off the stems, lifted the corms and kept them somewhere cool and dry in a paper bag, then re-potted them again in late spring. Watch out for weevil grubs in the soil.

If you are referring to tuberous begonias and they are in pots, once they finish flowering tip the pots onto their sides and leave to dry out. Once dry take off any remaining foliage (it usually breaks off quite easily), brush as much soil off the tuber as you can, and then leave for a week or so in a dry place (I put mine in a tray in a cold greenhouse). Once completely dry, brush off any remaining soil and dust each tuber with yellow sulphur powder (sold in garden centres). I then keep them in old washing up bowls covered in old dry compost under the greenhouse bench. If the weather gets really cold lay a couple of layers of fleece over the bowls.

The most important point is that they must be dry before storing or they will get mould.