When I have the time to go to the grocery store regularly, I spend about 30 dollars a week, at most. I am just a single 20 year old female that eats 3 meals and 3 snacks a day. 30 dollars is usually two trips, in which I get 2-3 large bags of fruit and veggies. I eat a lot of whole wheat pasta and oats for grains, and chicken breast for meat. It's when you start to stray into things like pre prepared sauces and mixes that you incur lots of expenses, but 30 dollars a week would not be unreasonable and would even allow for a few treats. Stocking up on meats when it's on sale also helps, and I find shopping and somewhere like No Frill's saves a lot of cash just by taking a trip down the sale aisle first and selecting low cost alternatives. I find fruits and veggies are very cheap, I can get a weeks worth of plums, nectarines, apples, oranges, banana, potatoes, etc, enough for 8 servings a day, for about 14 dollars a week. The other food groups make up the remainder. Buying what's in season also really helps - lots of squash, potato, and ground fruits and veggies in the winter, and tree fruits in the summer. I could easily live on 15 dollars a week by not allowing myself things like cheese, jam, popcorn, or cereal, but I do like to eat some amount of processed foods. Drinking water rather than juice will also help a lot, and base you dairy on what is on sale (milk, yogurt, or cheese - you don't need all 3).
Lately I have only been able to make it to the grocery store about 2 times a month, so I can't eat as much fresh food because it spoils. I have to eat out more, and with buying more frozen foods to keep longer, I'm spending closer to 80. I'd like to go back to my old ways but I just don't have time.

My family, on the other hand, a group of 3 (parents early 50s and 12 year old brother who is a light eater) have a grocery bill of roughly 250-350 a week. They buy a lot of junk food, cakes, fancy juices, pie, deli meat etc. They never eat fruit, and veggies are limited to what comes frozen in a bag. They also drink a lot of soft drinks, and their fridge is full of "fancy" food no one eats like 30 bottles of salad dressing, each which they buy for just one salad, 12 kinds of jam, etc - all things that are unnecessary. They buy specialty things for just one meal, like a cereal they have on bowl of and forget. It's things like buying hot dog buns AND hamburger buns AND bread AND rolls AND wraps that kill you, rather than simplifying and buying just one standard that will work for everything. Less waste, less overhead, more room in your fridge.

The cheapest way to shop is to base your diet on non-processed foods, and practice shopping around the outside of the store - fruits and veggies, rice, pasta, bread, whole cuts of meat (rather than processed burgers, breaded fish, or deli meats), milk, etc. The less processed food you buy, the cheaper your bill will be.

To have a reasonable diet where you allow yourself some amount of processed foods (french fries, cereal, etc), I'd plan for about 40-50 per week, per person, based on local food costs. A very clean diet could easily be 25/week/person or less, but when you get into eating little fresh foods and more processed, with takeout, something like 70/week is probably more reasonable.

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