There are many stories of dogs chasing cats and cats taunting dogs, but even more stories of dogs and cats living together, if not exactly as Best Buddies, then at least without enmity.
Many factors govern the relationship between dogs and cats, including breed, age, temperament, early socialisation and the number of dogs that form a pack.
To some degree both cats and dogs vie for the same resources; territory, food, shelter, access to humans, etc. The need is different and the needs of both species can be satisfied in one household.
Understanding the nature of both cats and dogs and their needs for safety is important, as is following the proper introduction process. This will be different for almost every situation so it is best to arrange for an animal behaviourist to help set up the correct plan for your pets.
Dog breeds that are of terrier or hunter origin are more likely to chase cats, although under the right circumstances any dog may chase a cat. If the cat runs, it looks like prey, a rat or a rabbit maybe, and the basic hunt instinct is triggered in most dogs although not all dogs will kill the ‘prey’.
Cats mainly run because they are in a vulnerable position or because they are frightened by a large, loud, ‘monster, or when they are threatened. Quite often the cat can outrun the dog and leap to a higher surface and safety. Cats that are cornered can inflict nasty scratches and bites when defending themselves.