The leaves could be old and will eventually dry up and fall off or you can remove them yourself. They could also have some fungal disease which can cause unsightly patches - remove these if you find them too unattractive.

As for sun exposure if you see burning on the leaves then you need to move them or provide some shade during the hottest hours. A lot of begonias can be acclimated to receive quite a bit of sun but not all day. There may be a few that can take sun all day such as Dragon Wings but they do need frequent watering.

What kind of begonias are they? Rex, tuberous, rhizomatous, cane, semperflorens?

Sad to say I'm not sure. They were planted as tubers so I don't know if that guarantees they're in the tuberous group? They are a cascading type of begonia. I wish I could be more helpful.

If it was a tuber you planted then it is a tuberous begonia. Either cut off the old leaf or trim away the browning edges (eventually the whole leaf will yellow then dry up and fall off if you don't remove it). This sounds like a humidity issue (too low but they grow beautiful tuberous in CA where the humidity is low). I've grown tuberous here but they are basically annuals due to our climate.

Here is a web site from Scotland where they grow magnificent tuberous begonias. They might have some answers to your questions.

Scottish Begonia Society

Thanks for the info.

Tuberous begonias need not be annuals. I grow the Non-stop tuberous begonia and save the tubers every year. They don't last forever but I've had them last for 5 years.

What to do:
After frost has nipped the plant but before the tuber can be frozen, lift the plant. Cut off the top leaving about 2 inches of the stem. Shake off as much dirt as you can place on a newspaper inside for a day or two to dry. Bury the tuber in peat moss in a covered container. Put the container in a cool dry dark place like a fruit cellar. Check it occasionally over the winter and give it a small spray of water if it looks like it is drying up too much, but mostly they don't need it

In the spring pot them up and give a little water--not too much--remember they don't have roots yet. gradually increase the water but never make them sopping wet. They take a long time to root, so start them about 3 months early. Until they show leaves they don't need sunlight just warmth. Then put them in a sunny window or under lights or in a greenhouse.