Ground covering plants or perennials are a very useful plant group. They cover the surface with their leaves or stems, reduce the drying out of the soil (it keeps the soil more microbiological, active and alive), they can prevent soil erosion, etc. What is more important, they can reduce the time spent on weeding. Nature doesn't often create monocultures, more often plants are mixed, so true ground covers are quite rare. There are plenty of creeping plants that cover the soil, but if they are not competitive enough, new tree or weed seedlings can occur any time (these usually work only in suitable habitat - usually sunny and drier soil). Simply don't rely on Ajuga, Thyme, Iberis or even native Phlox subulata and similar recommended creeping perennials, these won't do the work for you!
The good competing non-native ground covers for sun are some Alchemilla, Hemerocallis, Bergenia, Liriope, Stachys byzantina 'Big Ears', some hardy Geranium, Sedum telephium etc. For half shade and shade Lamium, Gallium odoratum, Brunnera macrophylla, Hosta, Geranium, Epimedium x perralchicum, E. versicolor etc.
With the native groundcovers we don't have a huge number of choices, but we still have some!