The difference between "rot" and "ferment" is extremely thin. The use of bacteria and fungus to partially break down foods is as old as any cultivation process. "Rotten" is just the word we use when the microorganisms have done something we don't like.

Fermented (rotten) foods are incredibly common, from cultured butter, ghee, buttermilk and yogurt to cheese, wine, soy sauce and beer. Not to mention bread, Tabasco, Sriracha, kim chi and sauerkraut, or heck, salami, chorizo, kombucha and fish sauce. Wikipedia has a huge list. We use bacteria like Lactobacillus, yeast like Saccharomyces cerevisiae, mold like Penicillium roqueforti and thousands more. Some are cultured specifically for their task, and others occur naturally and are allowed to do their business without interruption.

A few that come to mind that are less commonly considered and fairly interesting to me:

Bonito - Fish are dried, smoked and inoculated with mold (Aspergillus glaucus). A staple flavoring in Japanese cooking.

Kumis - A carbonated, fermented milk product. Lightly alcoholic, and traditionally made from mare's milk, and fermented with wild yeast and bacteria. Quite the beverage, I hear.

Hákarl

Iceland: Aged Greenland Shark.

According to Bizarre Foods with Andrew Zimmern, basically, the shark is cut into chunks of maybe around 10 lbs or so, put in a wooden box for a week or two, and then taken out of the box and hung for about 6 months in a wooden shack to essentially rot. Average temperatures in Iceland in the summer time only reaches about 50 degrees Fahrenheit.

The reason why the shark is allowed to rot is because if you eat the shark meat fresh, the ammonia in it's body is know to have cause others to spit up blood...basically it can do some major damage. The only way that they know to 'treat' the shark so that it is edible is to allow it to rot and I think let the ammonia leach out of the meat by hanging it to rot.

Century egg

Eggs are coated in alkali clay and burred for a few months. I have to say when I had the opportunity to try them I didn't. The fermented egg white is clear and dark brown, the yolk is a dark blue/green colour. The smell is very strong, mostly as they contain a lot of amonia.

Aging meat is part of everybody's culture. Freshly cut meat goes into 'rigor mortis' after about 24h and becomes tough. Letting it hang in a controlled environment (temperature and humidity) will let the natural occurring enzymes in the meat tenderize it.

Aging can last up to ... 120 days! But apparently that doesn't appeal to everybody.

Wikipedia

Kaanga Piro

Kaanga Piro is New Zealand Maori version of fermented corn

It's history is from the Maori wanting to preserve the late Summer/Autumn corn harvest for the winter. Corn from this season was often damp, and could not be stored as dried cord

The corn cobs are packed into kete (hand woven flax baskets), and submerged in fresh flowing streams. It will keep upwards of 6 months while submerged

After a few weeks in water, the corn cobs turn into to a mushy paste, and has a nutty taste and texture not dissimilar to hummus and fermented soybeans

It smells disgusting, but it is very edible

Expect to see it marketed globally in the near future as a novel food flavoring