When researchers at Cornell surveyed households that had added a second cat, about half reported fighting at the introduction, and in half of those the cats had simply been put in a room together. The households still fighting later were the ones where the first meeting went badly. That is the whole argument for doing it slowly: the first meeting is the one you do not get back.
The steps below are the feline practitioners' 2024 protocol and Ohio State's, which agree. The timeline is "a few days to a few months, and sometimes longer," and the one rule is: "if intercat tension occurs at any point in the introduction process, go back a step."
Signs to slow down, from the guideline: "urinating/defecating outside the litter box, not going near the item, growling, hissing, pacing, or changes to the cat's normal behaviors." A stare that does not break, a tail lashing, ears flat: end the session before it happens. And Cornell's warning: "never put your hand or any other body part between cats that are fighting." Throw a towel, make a noise, close a door.
In the long run, the feline environmental needs guideline's rule of separate resources is what keeps the peace: at least one litter box more than there are cats, food and water in more than one place, and somewhere each cat can be alone. 87.7% of multi-cat households in a 2020 survey reported signs of tension; most of it is about resources.
Puppies meeting an older dog: the older dog is allowed to say no. Supervise every interaction and give the adult somewhere the puppy cannot follow. If the new one is a puppy, its socialization window is closing by three months; the first-year timeline has the dates.
Ohio State's rule is that the cat needs a refuge the dog cannot enter, from the first day and forever. Gates that a cat can jump and a dog cannot, or a cat door. Then the dog on a leash in the room where the cat is, rewarded for calm, corrected for nothing more than a redirect: "the cat may hiss, swat, and growl at the dog. Do not punish her for this." And the line that is not negotiable: "you should never permit the dog to chase the cat or pick her up in his mouth." A dog that has chased once will try again; some dogs with a strong prey drive should not live with cats at all, and the honest thing is to find that out before you adopt.
No punishment, no "letting them sort it out," no holding one animal up to the other's face. The American Veterinary Society of Animal Behavior's position is that "only reward-based training methods are used for all dog training," because studies show "an association between the use of aversive training methods and long-term behavior problems including aggressive behavior towards people and other dogs." The same logic runs for cats: a scared animal learns that the other animal predicts something bad.
If it is still going badly after weeks of doing this properly, that is what veterinary behaviorists are for, and a referral costs less than rehoming.