Brown edges are an indication that the plant has been kept too dry or the temperature is too high, or both. General yellowing of inner leaves indicates over-watering. Unfortunately, this is a virus that is fatal and very contagious. If your begonia develops yellow spots and the leaves begin to brown along the edges, put a sample in a sealed zip-lock bag and have an expert look at it. Begonias with mosaic virus should be destroyed.
Leaves tend to drop when the plant is exposed to cold temperatures.
Some varieties of fancy-leaf begonias need to go dormant occasionally. Try setting it aside to rest for 4-6 weeks. Then water it and put it back in good light. It should develop new leaves. White or gray patches on the leaves are most likely a fungus.