The risk is usually low, and specifics vary from food to food. But when in doubt, throw it out. Since food packaging can carry different dates (use by, sell by, and best if used before), knowing when an item has gone bad can be tricky.
With milk, for instance, keeping the refrigerator adequately cold and discouraging carton slurping (which introduces bacteria) are more important than the date. With deli meats, however, the date is key. Sliced turkey or ham that has been around too long (more than two weeks past the date with a sealed package, or three to five days after opening) can be contaminated and, in the worst case, cause a potentially lethal bacterial infection called listeriosis. Packaged foods eaten after their expiration dates will probably just taste bad, not harm you.