This may not work for everyone, but my favorite is strawberries. They can get a little thick and matted after a while, but if you're growing them as a groundcover and picking the fruit is secondary, it's not that big a deal.

For shady areas, I like sweet woodruff, but it can also get a little carried away :)

Any fast growing groundcover will present constant problems of controlling it. Try mulch to control weeds.

Thyme is a great ground cover