Sounds like too much water. Has it rained a lot lately? If the soils is not porous, begonias will drown. Use a lightweight soil. You can also spray with fungicide, but the roots need to dry out. Greg Sytch ABS Horticultural Correspondent

Help- the leaves are getting a brown center and are falling off the plant when the brown area is about 1 cm in size. Otherwise the plant looks healthy and I do not see any insects.

Are they indoors now? I wouldn't worry too much about them during winter - just give them the best light you can. Most important thing is to keep the stems firm and healthy. Pick off the old, diseased leaves and new leaves should appear if conditions are good for growing.

Your first description sounded like some powdery mildew which can be hard on certain begonias and just a pest on some others. Spray with a fungicide to keep it under control.