When it comes to dealing with aggressive animals, it helps to understand the reasons why animals become aggressive.  While certain breeds of dogs have been bred to be aggressive or overly protective guards, most animals are usually more aggressive when it comes to their own territory.  Both cats and dogs are territorial animals and will stake out their own territory and defend it.  This usually works in two ways- one, they must work out territory amongst themselves, and two, you must be aware of territorial concerns when entering the properties of others.

In-home territorial disputes often happen in the home when a new animal is introduced.  Consider your existing pets carefully when bringing a new animal home.  For cats, most experts say that a younger, smaller cat is the best addition to a household unless you have geriatric cats.  The longer a cat stays with its siblings in the litter, the more sociable it will be with other cats.  You can assist cats in behaving nicely toward one another up to a point, but territory is something they will suss out on their own.  Here's a trick for if you find that when one cat returns from the vet the other begins to behave aggressively toward it: rub the liquid from canned tuna on your hands and then pet both cats.  Not only will they both smell the same to each other, but this will inspire grooming in both animals, perhaps even of each other, which is very calming for them.

Dealing with other people's animals means respecting territory.  This applies mostly to dogs, especially those who are guarding not just their physical territory, but the rest of their pack, which includes their human companions.  If you enter a space only to discover there is an aggressive dog present, stop, stand still, and do not make eye contact.  If the dog remains still, you can begin to slowly back away, never turning your back on the animal, and never moving quickly, as this can trigger them to chase and attack.  If the dog does attack, your best bet is to find something to throw at the dog, or offer out a purse, briefcase, or other object that you can then hold on to, giving you extra time to plan your exit.