How can I fix a beautiful concrete driveway that floods in heavy rain?

Photo? Without seeing it, when poured it may not have had enough slope or pitch for run off. Instead of ripping up expensive concrete, I would install French drains on either side of the driveway to direct the runoff into the street and into storm drains.

Naomie is right we really need a picture to give you a good answer.

Hi! Redirecting the water is indeed your best option. Check where it is coming from. Some redirection of gutters may help.

Check out the "Hydro Blox" system. If you rent a trencher, you can install it yourself. Longer lasting than French drain, and capable of handling more water.