I've gone through biting and hitting phases with my 3-year old. First thing you should know is that this is common and not your fault, though very frustrating. I did pretty much all of the things recommended below and mostly it just took took time, repetition and a lot of patience. You may be getting funny looks from other parents who haven't experienced this yet -- they are fortunate not to have gone through this and should just be thankful.
One thing that made an impression on my son was completely leaving a situation (playground, party) if he bit or hit. I would explain appropriate behavior before we went into whatever, remind him that if he felt frustrated or couldn't control his body he could come have quiet time with me or ask to have a time out (to cool out), and that if he bit or hit we would leave. Yes, this did take several times of actually doing it for him to understand, and yes, he was as young as your daughter for the biting part. And I do still remind him about expecatations for his behavior when we go anywhere.
I've also been careful about when and where I take him. He does better at outside playdates or places where there is lots of space and lots of toy/activity options; I arrive early for parties so he can settle in before it gets too busy, and I watch him closely for signs that he is getting wound up and may have trouble controlling his impulses. Then I either take him away for a quiet activity together (sticker books, reading, walk outside, snack) or say thanks and goodbye -- get while the getting is good. I don't take him places near naptimes or in the evening, and I make sure he's well fed anywhere we go.
Yes, this is very high maintainace, but it has gradually helped him play better with other children, learn boundaries and kept us sane.
Some good books: for your daughter "Hands Are Not for Hitting"; for you "Raising Your Spirited Child." The second book may not apply, but if your gal is intense in other ways this is a very helpful book.
Hang in there. Best wishes, L.