In response to a post above which - disturbingly - went unchallenged, Roundup has been found by studies to cause cancer. The World Health Organization’s (WHO’s) cancer authorities – the International Agency for Research on Cancer (IARC) – determined that glyphosate is carcinogenic to humans. Exposure to 2,4-D and dicamba is linked to non-Hodgkin’s lymphoma, the same cancer with which glyphosate has been associated. Herbicide exposure in general is also linked to increased rates of Parkinson’s disease. The IARC is the world's leading authority on cancer, and unlike many other organizations, has no financial pressure from corporations to fudge their numbers or turn a blind eye to sound medical evidence and replicated studies. Additionally, massive use of glyphosate with Roundup Ready crops has generated an epidemic of glyphosate-resistant weeds, and the development of GE crops resistant to herbicides such as 2,4-D and dicamba is in turn generating an unprecedented and increasingly toxic explosion of weed resistance and herbicide use in US agriculture. If you don't want to give your family cancer and Parkinson's, and increase the number of super-weeds in the world like overuse of antibiotics increases super-viruses, don't use them! Weeds, after all, are not going to kill you. Weed killer, on the other hand, will.
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There are a lot of things to try in addition to those already mentioned here. First, for small areas to do by hand, I have found that a Hula Hoe is very easy - you can use it standing up, moving it back and forth sort of like a vacuum, and it pulls the weeds right out. For more stubborn weeds, you can sharpen one edge of it and use it to hack roots very sharply - I have beat the hell out of certain tough weeds and small invasive trees, and the Hula Hoe is none the worse for wear, but the weeds sure are!
Another option is to use Australian Pine needles - these are not actually pines, and are invasive in Florida and some other areas with similar climate. If you happen to have access to them, they are easy to get in large quantity, and the reason they are so terrible is that the needles are notorious for making it difficult for nearly anything to grow through them. They are not actually pine though, so unfortunately you can't use them in composting to enhance the acidity of soil - but you CAN use them as very effective mulching for already-established plants, and usually people are only too happy to have someone truck out the huge pile of needles (actually tiny green branches, not needles) in their yard.
You can also use intentional ground covers that are likely to choke out the weeds, or make them invisible. There are quite a few nitrogen-fixing options you could choose if you want to help bordering plants. You could also choose to keep weeds that use minimal resources and look attractive, or are useful in their own right, and get rid of the other species.
Another option is to put hydrogen peroxide in a spray bottle and spray very directly onto only the plants you want to be rid of - there is no harmful effect on the environment, and with 1 or 2 applications the weed will die. You can do the same with vinegar. But don't do this too close to the roots of plants you want to keep!
Also, make sure to identify any "weed" before trying to remove it. A lot of plants traditionally identified as "weeds" are actually more beneficial as food, and in some cases tastier, than the plants you're growing intentionally! For example, purslane, amaranth, dandelion, plantain, chickweed, and dollarweed are all examples of EXTREMELY nutritious (and often delicious!) plants that you may be so lucky as to have growing for free. Don't kill yourself ripping them out by hand, or pouring poisons all over where you live - just go out into your yard when it's time for a meal and gather some fresh greens to sautee or make a salad with! Many of these have very strong medicinal properties, are anti-inflammatory, anti-cancer, and can really boost your immune system as well. Many can even out blood sugar and manage diabetes, or in the case of plants like amaranth provide a complete protein in a large enough quantity to replace expensive meat. There is a long list of the "weeds" in Florida, for example, that are edible, and are far better for you and fresher than the things you throw your money away on at the grocery store. So carefully identify the plant FIRST, before you throw away the thing that could possibly save you from a heart attack, or costly trips to the doctor to get Rxs to manage chronic diseases you could have managed, with the free plant you were hell-bent on eliminating from your yard.
I don't think it's worth obsessing over ridding the yard of ALL unintentional growers. Keep in mind, plants fruit just fine in the wild with plenty of competition, and to a certain extent, a little grab for resources encourages them to be stronger and more resilient. They have to work for their food so they don't get complacent. Not to the point of letting them get starved out, of course, but a few weeds here and there are not going to stop a healthy tree or shrub from fruiting.
There original post, which was a few years ago, mentioned shade cover, and I would just note that shade does not mean one will eliminate weeds. There are different types of weeds that like different conditions. Outside of invasives, there is no real definition of the word weed - it's just a plant that you didn't intend to grow. Some of them are far more beneficial than you realize either to the garden, to getting rid of certain pests, or as additional food or medicine. An unintentional plant can grow in shade, in sun, in bogs, in desert... plants grow everywhere - even in rocks! So shade is not going to stop plants from growing, not at all!