Weed free ground covers?

I have several areas under trees - part sun - that I would prefer not weeding.  I am experimenting with ground covers as a way to suppress weeds.  They are surrounded by grass so I can mow around them and not be concerned about very vigorous ground covers spreading where I don't want them.  I expect to pull out some poplar/maple/locust sprouts every year but other than that are there ground covers that will totally take over and suppress almost all other weeds?

Experimenting with hostas, sort tooth mountain mint and Lilly of the valley.  Was thinking of trying hidcote comfrey.  But I know it will take a couple years for them to get established and know how well they are doing.

Anyone have something they use that really works?  

Some people try to fight it, but I welcome Glechoma hederacea a.k.a. ground-ivy a.k.a. gill-over-the-ground a.k.a. creeping charlie a.k.a. alehoof a.k.a. tunhoof a.k.a.  catsfoot a.k.a. field balm and run-away-robin. It stays low to the ground, out shades everything else, and has nice little flowers that the bees seem to love. I tried introducing dutch white clover due to its nitrogen fixing goodness, but Glechoma hederacea is much better at out-competing other things.

In Maryland pachysandra should spread easily and is capable of crowding out weeds anywhere you don't mow, while not spreading to mowed area.

I'd agree with pachysandra for Marykand area. When I lived in NJ, I had pachysandra and it was an easy ground cover.