In Formula 1, yes they do race in the rain, even when it is coming down in buckets. They have monsoon tires, wet tires, intermediates, and dries, as well as hard and soft compounds. If the rain is insanely excessive, however, they can red flag a race (this has not happened since I began watching). Some drivers are good in the rain, some aren't... yes, it takes a lot of skill to drive while your car aquaplanes (they say aquaplane instead of hydroplane in F1).
Driven is based on CART. Interestingly enough, I do not believe they race in the rain (although I could be wrong... I doubt they race in the rain on ovals).