To prevent both weeds AND erosion, ground covers need to cover the ground all year. So, true ground covers are evergreen. 

Sun-loving Creeping Junipers are great; some are native to our region.
Many Sedums like this one make great groundcovers in the sun. This is S. takesimense.
Carex ‘Ice Dance’ fills in nice and thick. Virtually carefree, unless it becomes ratty in late winter and needs trimming back. Can take full shade, or sun with supplemental water until it’s settled in.
Another Carex (C. morrowii) in full shade.
Liriope spicata fills in fast. Ok in sun or shade.
Mondo grass is very short, can handle shade and a bit of foot traffic, but fills in very slowly.
Mondo grass replacing asphalt/cement under cars in a demonstration garden. (At River Farm in Alexandria).
Packera aurea or Golden Groundsell is a great native ground cover for shade. Spreads quickly, blooms in April.
Ajuga reptans (center and foreground) can take sun or shade, but is often short-lived. Here it’s shown with Carex ‘Ice Dance’.
Comfrey fills in pretty quickly and blooms in April. A shade-lover, it needs extra watering if in the sun.
Periwinkle covering a slope in GHI where, surrounded by asphalt and concrete, it can’t spread anywhere. or up into the tree It’s on some invasive-plant lists, so it’s forbidden to PLANT it in GHI,.
Pachysandra also stays on the ground , and can take a lot of shade.
Native Pachysandra fills in much more slowly.
English ivy is a true menace! It climbs trees to potentially harm them, and while there it matures to a different shape and makes berries, which birds then spread all over the place. Invasive to the max! Seen here in GHI, with berries ready to be spread.