Put these unwelcome guests to work and let them tell you what's wrong with your garden.
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Your soil might be too wet for the plants you're trying to grow but not for these weeds. Here are some examples of weeds that thrive in wet soil:
Once weeds become established, many can adapt to their changing environments so fixing the problem won't necessarily kill them off. You need to remove them and do it carefully so they won't spread. Make sure to get all of the seeds or tubers -- anything they can reproduce with. Yellow nutsedge is particularly difficult to remove because each tuber will sprout a new plant; it's very aggressive in the right environment.
To amend the problem, you'll need to fix your drainage. There are a lot of plants that don't like soggy soil, so if you can get that water to drain away, they'll start looking better. Check your drainage system to make sure it's not clogged and amend your soil so water flows through it better. Fixing the problem may even be as easy as cutting down on watering.