Everyone has her/his/their own group of invasive species with which they have to deal (though I suspect that some, like multiflora rose [Rosa multiflora] and Japanese honeysuckle [Lonicera japonica] are pretty ubiquitous, at least on the East Coast.
In the natural area that I frequent, the most troublesome of many, many invasives is porcelainberry (Ampelopsis brevipedunculata). Porcelainberry is a member of the Vitaceae, and there is even a native species found on the Coastal Plain of the Carolinas. But this plant is definitely not the native species. The folks at the natural area call it the "kudzu of the North," and you can see why.
Porcelainberry was introduced as an ornamental (you can see why, from the image of the fruit), and it escaped from gardens. The fruit are readily taken (and spread) by birds. I had a friend who worked at the Berkshire Garden Center. The organization had a variegated specimen of porcelainberry espaliered on one of their buildings. In horror, I told him that he'd better get rid of that plant immediately, but he said that it didn't spread in that climate. I wonder if he was right.
Porcelainberry develops a very thick root that stores lots of reserves; it's virtually impossible to kill it by starving it to death with persistent and diligent cutting. Periodically, it sends up an stem from the roots, so that the plant spreads both by root sprouts and by seed. It's a monster, and be happy if you don't have it.
Mile-A-Minute (Polygonum perfoliatum)
And, while we're on invasives, the second biggest problem in my natural area is mile-a-minute, a.k.a. Devil's tail and tearthumb (Polygonum perfoliatum). This invasive was introduced on nursery stock imported from China to York County, Pennsylvania in the 1940s. It has infested the Mid-Atlantic.
This plant is an annual. Unlike porcelainberry, which adds to last year's growth, mile-a-minute must start over each spring. It produces copious quantities of blue fruits that are spread by birds. Fortunately, the managers of my natural area have introduced a biocontrol--an imported weevil (Rhinoncomimus latipes)--that, for now anyway, is specific to mile-a-minute. The minuscule weevil larvae burrow into the plant's stems and the adults perforate the leaves (as you can see in this image). The beetles have only been around for two years and haven't yet made much of a difference, but their numbers need to build up to be effective.
Persimmon fruit (Diospyros virginiana)
On a walk yesterday afternoon, I came upon two persimmon trees loaded with fruit. The ground around the trees was festooned with the mushy fruit as well. I don't particularly like persimmons, but they're good for a zeitgeist: they remind me that it's autumn.
I love the tree's blocky, dark-gray bark during any season of the year.