The easiest answer is to always be consistent in curbing aggression. Never let your cat scratch or bite. If you're playing, and this starts, say NO, and make your hand or foot go limp. Slowly extract yourself from the cat and ignore her. You can give her a soft bop on the nose - not hitting, but picture the way mom cats use their paws to tell their kids that enough is enough.
As a side note - corporal punishment is unacceptable for cats. They do not respond to this type of negative reinforcement - they just don't. Hitting your cat will do nothing but make your cat terrified of you. They do not associate being hit with a behavior - they just associate it with you.
Now, try to figure out if it is only certain situations that give rise to the aggression. If so, you can obviously change those situations. If the cat attacks you every time you reach across the table for the remote, maybe start leaving it on the couch. I know, it's a stopgap, but if the answer is simple, just use it. It's less stressful for everyone.
Is it directed at one person, or at everyone? If it's one individual, that person has to encourage a new relationship. For some reason, the cat is afraid of them, or sees them as a threat. They have to make sure all their interactions with the cat are positive, to make the cat associate them with good things. Make sure they speak softly around the cat, give him treats, and become the primary caregiver (dole out the food).
If it's directed at everyone - you may have a power struggle going on. Your cat is trying to become the dominant member of the household, and this creates serious problems. You have to take everything away from that cat - toys, food bowls, everything. Then be very controlled with how he gets them back. The cat is fed when YOU say he is. He gets his favorite toy only when YOU say he can, and you take it away when YOU think he's had enough exercise, or starts becoming aggressive. This is a quick way to put cats back in their place, and when they realize they have no way to be in charge and you've reestablished your dominant position, the behavior should be gone.
Is your pet a solo pet? You may not be giving it the play it needs to get all that energy out of its system. Try to target the situation and distract the cat. For example, if you get attacked each morning after you get out of bed like I currently experience, put a stash of toys on your nightstand. Then, first thing, start a play session. And you can't just pick a toy up off the floor - that's not going to be interesting or special.
Do you have a new pet? Hissing, growling and spats must be expected. Just keep them separated when you can't supervise, or if it gets too rough. Screaming and real violence are not good signs. That is not to say that it can't work out - you're just going to have to go slowly. Expect setbacks. When they happen, reconfine the animals. This kind of violence can develop between old friends. Something went wrong between them, maybe one came home smelling like the vet's office after a stay - and now, you must treat them as if they're newly introduced cats.
An important point - if you bring home a new cat, ask a friend or a neighbor to actually bring the new cat inside. Your cat seeing you bring in the new guy and open the carrier can lead to his resentment of you, and could lead to attacks. Cats often redirect attention - they're too scared of the new guy, so they go after the owner because they know the owner won't hurt them. Don't get too upset if this happens to you. Just help the cat stay calm, and be patient.
Stick to a confinement schedule for at least a week, and slowly introduce the animals. Keep one in one room, one in another, and start to feed them near the doorway. This way, they can catch each other's scent while getting food, which puts a positive spin on things. Then, begin to switch rooms, and allow them to explore and sniff the other cat's room.
If the cats don't come around, and one (or both) remain aggressive toward one another, you may have to consider medication. This will include medication for the aggressor and the victim, who will likely be so traumatized by this time that they'll need assistance with getting over the anxiety the aggressor causes them.
