Food expenses have been a major area I've cut back on. Back in the US, my husband and I spent ~$50/wk on food ($60/wk when we were renting a place without an oven). When we moved to New Zealand for work, our bill was initially NZ$80-100/wk (US$58-72), but we've brought it down to NZ$20-40 (US$14-29) per week. My initial impetus was to cut down our budget by NZ$20/wk so that I could afford to buy tools (especially woodworking tools) for my many projects. Here are some things we've done to keep costs down:

• Reduce the amount of meat we eat. Growing up, I ate a more standard American diet, with at least one meal per day (and usually two) containing meat, approximately 100lbs per person per year. I cut down to ~50-100lbs per person per year when I left the nest, but since moving abroad and getting pinched by high cost of living, it went down to ~5-12kgs (11-26lbs) per person per year. I sometimes supplement this with other (free) meat from hunting, hunter friends, self-raised animals, and roadkill. Not being picky and knowing how to slaughter/butcher animals helps a lot. Cutting what meat you do have with lentils also helps, as well as using meat as a seasoning rather than a main course.
• Cook from absolute scratch where possible. Processed food is almost always more expensive than making it yourself. We make our own bread, pasta, bagels, pizza crust, puff pastry, phyllo dough, you name it. This allows us to have luxury goods, like apple strudel, for cheap. It also limits our ability to overeat, as we may want more dessert, but not enough to make a second batch! One area we have saved a lot of money (apart from cutting back on meat) is breakfast; cooked rolled oats with salt and a fried egg keeps me going until lunch and costs under 60c (US40c) per person (and would cost less in the US due to lower food prices). I went into the cereal isle last week after a three-year absence and was shocked at how high the prices are--I am glad I quit processed cereal a decade ago!
• Eat less. Most people these days eat too much and get too little nutrition. By eating smaller meals, higher quality ingredients become more affordable (as you are using less of them). I typically eat a single, small bowl of food for dinner, and it is enough; to prevent myself from being hungry all the time, I make sure that my meals have a high fat to carb ratio. This diet has not negatively impacted my health; I'm fitter than I've ever been in my life and my BMI is a perfect 20.
• Eat produce that's in season. Locally, new potatoes are available in early summer, and aged potatoes in the late summer and autumn. Cabbage is cheaper in the autumn here, as are pumpkins. Beetroot is available year-round. I harvest as much fruit as possible from the local area, store up what I can, and only pay for fresh fruit in the late winter and early spring--but this is a luxury.
• Grow the expensive vegetables at home. In the past three years, I've bought one head of lettuce, and no other leafy greens. What greens I eat, for the most part, is what comes out of my garden (I am in a temperate area and can grow brassicas in the winter). Lettuce, silverbeet (chard), basil, parsley, kale, spinach, New Zealand spinach, artichokes, rhubarb and herbs are things I grow and don't buy at all. I have started many, many gardens in the various places I have rented.
• Keep a pantry of staple goods. I always have a 20kg (44lbs) sack of white flour, wholemeal flour, and rice on-hand. I don't own a car, but it is possible to carry these in a camping backpack to bring home. I can all the jams and tomato sauce I'll use in a year, and some of the pickles. I would like to get back into canning fruit in syrup, but have been limited by space and availability of jars. I keep quantities of spices (some store-bought, some dried from the garden), herbal teas (self-harvested mostly), and other goods like butter and oatmeal in bulk as well. I typically have ~2 months of food on-hand. Not running out of staples, and having a menu that uses those staples, prevents costly last-minute food purchases.
• Shop around. I agree with dtrammel and the article on the importance of this point. An example: the price of butter is on the rise in New Zealand just now. It could be obtained for $2.50-3.50/500g as little as four months ago. Now the lowest price is $5.00/500g. However, there are places selling it for $5.50 and even up to $6.50 for the exact same product. I frequent ~13 different grocery facilities in town throughout the year, making 0-3 grocery trips per week (average is probably 1). The farthest is 22k (15mi) away, which is ~1 hour each way by bike; I typically only go there if I have other errands out that direction. The cheap/free pork leaf fat made the trip very worthwhile over the years.

All that being said, probably 1/4 of my food budget goes to chocolate and butter--don't feel that I'm living a life of asceticism! I'm interested to hear other folks' tips and tricks.

It's easy to find, using Google, and claims that a US adult should be able to meet nutritional needs (as specified in the footnotes) on about $200/month per person. They've compiled a booklet of meal plans and recipes with a two-week cycle which includes variety, and economy of cooking (i.e., cook enough for provide leftovers for the next day).

I've seen the USDA Thrifty plan.
If you work at it (and I do), you can spend less than 200 per person per month.
I spend about that much for a household of four adults.
HOWEVER, the USDA thrifty plan food budget doesn't include pet food and mine does.
I don't think it includes paper goods, laundry soap, toiletries, health & beauty, etc. and mine does.
The key is stocking up when items are on sale.
Keep track of prices and buy only when the price is right and then buy enough to last your household until the next sale.
I don't use many coupons. Most of them are for highly processed foods I never use.

Also, learn to cook.

I devote a LONG chapter to grocery shopping in my book 'Suburban Stockade' that David was kind enough to recommend.

I will also say that it's darn hard to cut the USDA numbers down to say, $100 per person per month, without involving other people's money. You can't make any mistakes, you shop rock-bottom, and you don't waste a single thing.

Teresa from Hershey