Some people consider horseherb (Calyptocarpus vialis) a weed, but I think it's an amazing groundcover. It thrives in sun or shade (although it's not crazy about full sun all the time), doesn't grow tall but stands up to being mowed or trampled on, transplants easily, and doesn't require much water. Its springtime little yellow flowers are pretty, too, which is one of the reasons I've chosen so many other purple-blooming plants for contrast. Also, bees and butterflies seem particularly attracted to blue/purple blooms. A good thing!
We have worked long and hard to create a xeriscaped front yard--it's more like a huge drought-tolerant garden than a typical front yard--with no grass to mow, and horseherb is our primary groundcover. We're now working on spreading the horseherb in the back yard near the veggie beds, too. Someday there will be no pesky grass back there either.