Ground elder (or ground-elder, Aegopodium podagraria) was once widely planted as food and medicine (for arthritis and gout - it is also known as gout-weed) and this is the main reason for its wide geographic distribution. After a while, though, gardeners cottoned on to the fact that once you don't want it any more, it lingers on as an astonishingly tenacious weed, and has an upsettingly strong tendency to invade new areas. For the last couple of centuries or so it's been seen as little else but a weed. It's hard to prove, but it's quite possible that many of the patches of ground afflicted by it to this day were first planted deliberately some time around the middle ages.
The plant maintains extensive root systems, and spreads mainly by means of underground rhizomes. It is said that even tiny fragments remaining in the soil will quickly grow back into full-fledged plants if they are not dealt with, and once you have a plant, you'll soon have a whole patch.
Still, it could be worse. Ground elder - a member of the carrot family - is actually pretty tasty, and one way to deal with an unwanted patch in your garden is to just keep on eating it. After all, no plant can survive indefinitely without photosynthesising - many gardeners recommend covering ground elder with opaque plastic for a really long time to get rid of it, which strikes me as a wasteful and ugly solution, but it would certainly save some effort. If it's a lawn that it's infesting, regular mowing should eventually starve it.
The leaves are particularly pleasant to eat when they are young, and the leaves are bright and waxy. At this age they are quite tender, and the flavour is mild. They make a fine addition to salad, though for my part I often just tear the shoots out of the ground and eat them on the spot. They come back after about a week, so you'll need to be harvesting pretty regularly.
Ground elder is still quite edible when it's a bit bigger, and can substitute for spinach in many recipes. It has an interesting flavour that I'm at a loss to describe, which may not be to everyone's taste. I find it's delicious steamed, with a bit of salt and pepper and some nice oil (some might prefer butter). My favourite ground elder recipe so far is probably ground elder pakoras - just blend or chop a bunch of leaves, and mix with gram flour batter spiced with cumin, coriander, turmeric and what-have-you, then make patties and fry them in plenty of oil.
The plant starts flowering in late spring or early summer, and apparently the leaves start to have a laxative effect after this, as well as an overpowering flavour that most people find unpleasant. I haven't really experimented to confirm this. The flowers come in pretty white umbels, much like the flowers of the true elder. The two plants are not at all closely related, although the leaves are also superficially similar.
Last year I decided it was nice having a ready source of free leafy greens in my back garden, and made a conscious decision not to kill off my ground elder patch entirely. I'm not sure I could have even if I'd tried, though. This year I might be a bit more ruthless, and try to reclaim more of that ground for other plants, but I'm quite looking forward to eating my way through whatever the ground elder patch throws at me.