How do they shoot fight scenes without causing harm to performers?

The fight is shoot in many small pieces,in movies you can see how camera always changes many positions during fight scenes.

The camera position faces so that the fist blocks the view of impact,actors keep the distance between them so actually there is no real contact between them,the sound and the impact effects are added later on.Each scene is reviewed to make sure that actor that gets hitted timed accordingly to the first actor that is hitting him so that it looks believable to a viewer.

Also actors can perform parts of the fight in slow motion to make the best timed hits and other complicated awesome moves,it is edited and fixed later on so it looks like real time.

In movies like matrix with longer and steady camera scenes the fight requires a lot of choreography from actors and from the cameraman and it includes a lot of extras and ropes.

The moves are all scripted, I believe; so every actor knows which hits are coming up next and it's not a surprise...unless it's improvised like a slap or something (in which case that is totally real).

stunt men for the bigger action bits. Then, after they're on the ground and wounded, having just jumped down a few stories, put in the 'realies' (the actual actors, the faces people paid to see), put blood on them and shoot them standing up and talking. It's all choreographed, rehearsed and well-worked out. Action actors know how to throw and take a punch. Some, like Keannu Reeves are very skilled athletes, have a coterie of stunt men they trust, and will do things themselves when possible.

For the old John Wayne fights, they coordinated a swung punch just missing with the recipient moving their head quickly as if they were hit. Sound effects fill the credibility gap.

Sometimes in comedy shows, the person being hit will move their head the wrong way.

In modern digital action scenes, there can be wires, harness and such which are removed from the image later.

Also an actors face can be digitally added to a stunt performer. In Titanic, in the scene where our heroes run away from the gushing water, there was digital face replacement.

Whenever characters run from an explosion or fireball, it is always chromakey. It's actually easier to do it that way as well as safer.

Don't forget, there is such a complex set of insurance, Producers can't put a star at risk without the insurance company approving, otherwise a dead or injured star costing a hundred million or more might come out of the Producers own pocket. Double ouch!