What are some challenges that authors face when writing believable historical fiction?

The biggest challenge is getting the history right. You really have to do your research to be sure that something happened at a particular time, that certain people were at that place at that time. You have to have the little details correct too -- what sorts of furnishings were in the houses, did they use gas or electricity, what did people wear -- there are a ton of little things you have to get right or your readers will pick up on them and complain. You also have to take language into account -- make sure the Words you want your characters to say were actually used in common language back in that time!