Usually, I'm singing the praises of native plants, and these days, any plants invited to reside in the yard--besides things like herbs, vegetables and potted annual flowers--are native to the Chicago region. I love native plants: they are exciting in themselves and in how they strengthen ecosystem connections. However, plenty of non-natives stick around, plants that came with the house, or as presents or that I put in long ago when I thought gardens were supposed to be primarily non-native and as English as possible.
Because my general attitude is not to harm living plants if they fit in with the garden ecosystem, I don't entirely sympathize with those who, after their natives-only epiphany, go and rip out all the offending exotics and completely redo the whole garden--it's a little too baby and bathwater for me. Better to evaluate on a case by case basis, and to make substitutions as opportunity occurs. For example, a climbing rose I loved died of rose rosette disease. Naturally I wasn't going to put in another hybrid rose, especially in that spot--but now I'm looking for a place to put a prairie, or Illinois, rose. In late summer or early fall you'll see me smother mulching a little more grass to make room for more native shrubs, forbs and grasses (described
here).
Yes, I do draw the line at invasive species, many of which are dangerous to native ecosystems and plant communities--to have bishop's goutweed is to battle it, to see garlic mustard is to pull it out, and buckthorn exists to be chopped. In my garden, gone are some euonymous bushes, and I'm still working (forever) on the goutweed, old-fashioned orange daylilies, lily of the valley and English ivy. Unfortunately, many growers still produce and nurseries still sell all kinds of plants they just plain shouldn't. And don't forget, some natives can also wreak havoc in the garden: no old field goldenrod for me, thank you very much.
Yet there are plenty of non-natives that I'm not such a purist as to take out just 'cuz, though I wouldn't buy them now.