Mine has been hitting and biting me since about 18 months, but it's getting worse. He'll be three this fall.
I don't punish, but I do give natural consequences. The natural consequence of hitting or biting someone is usually that that person won't want to play anymore. So when he hits or bites, I immediately put him down and we end whatever game we were playing. I tell him (as calmly as possible because sometimes it really hurts) that I don't like to be hit or bitten and if he does it again, I'm going to put him down or end the game or whatever. If he does it again, I follow through immediately. It's not a time-out because I don't stick him in a room by himself or anything, but that particular play session is over for awhile.
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If we're out somewhere where I can't put him down, or if he's just melting down and really can't hear me, I'm not above gently restraining his hands and helping him to control himself. I tell him that I'm helping him control his hands and I realize he's upset, but that doesn't mean he gets to hurt me. I wish I could say this is always successful, but it isn't. I don't know if what I'm doing is the right thing but it seems to work more often than not. I notice that he lashes out physically when he feels unheard or misunderstood, so I try to get to the bottom of what's wrong. Sometimes though it seems like he just needs to get his big feelings out. I try to honor that, but I won't let him beat me up either. It's hard.
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Maybe someone with more experience can chime in. I'd love to hear some new ideas.