Not sure what is wrong with it other than it has something out of balance. If your stems and roots are healthy, remove the offending leaves and it should replace those in a short while. Often this is good for the health of the plant (I removed all the leaves of a small hydrangea last year due to spider mites or some other malady - it bushed out in a few weeks as though nothing had ever happened). If the roots are rotting then take some cuttings and start it over. Fresh potting mix often makes a big difference too.

I think a photo of the whole plant would help better

Thanks for the responses, I'm almost afraid it's something worse because many of my other begonias are affected as well. Teen, I would post a photo of the whole plant, but today I pruned nearly all of my begonias down to 3-4" from the soil. I'm going to give them a chance to come back out, and if the new leaves are not healthy, I guess I'm going to have to get rid of all of them. So this is the only other photo I have, which was taken before the pruning. I have a big cane nearly 4 feet tall, which only has leaves on the tip. I did not prune that one, because for the moment it's leaves look OK. I really to hate to lose all these, they were absolutely beautiful last fall.

Don't give up that easily. Cut them back, start the tops over, give them a good growing spot and water regularly. You might be surprised on how well they respond.