My DD1 was like this. I simply couldn't believe the violence of her tantrums. Screaming, kicking, hitting. For 6 months or so I found it impossible to deal with and was getting nowhere (except exhausted, furiously angry and depressed). I then got a bit of a better grip on myself and was more consistent. When at home I physically picked her up (trying my best to avoid the kicks and punches) and put her on her bed with her comfort blanket and teddy then left and shut the door. She was liable to trash the room and scream. After a few mins (maybe up to 10 mins) I could generally hear that she was calming down. I'd go in and ask if she was ready for a hug / to talk. If she continued screaming and thrashing around I would go again, and come back 5 mins later. Once it was over I would give her a hug and try not to make a big deal of it. If there was something that needed to be talked about (e.g. if the tantrum had been sparked by her doing something naughty), I found it was generally best to discuss it a little while later.
I really wouldn't try too much reasoning with her (even though she is v articulate etc for her age), once in the throws of a tantrum all rationality is lost, so even if she can articulate what is wrong she is unlikely to respond positively to a logical answer or reasoning.
When out and about it was hell. I cannot count the times I had to struggle a tantrumming 2 yr old plus baby onto the bus and somehow get us all home. Be strict. First sign of a tantrum - leave, whether you are at a party, a friends house or a park. She will soon get the message not to behave like that. If shopping or doing something vital that you can't leave, I found that just stopping helped. Stop (in the middle of the supermarket), stand still and just wait. Don't reason or cajol or anything. Just stand there (I know - sounds odd!), but it seemed to help. When you see a chink in the tantrum say firmly that you have to finish this and the sooner you do, the sooner you can all go home, if she starts screaming again then continue to stand still and silent. Don't worry about nasty looks from strangers, in my experience the vast majority of people who see someone with a tantrumming toddler are supportive and wish that they could help or at least would give me a little smile of solidarity and support!
Oh - and sometimes, depending on what had kicked it all off and how bad she was, sometimes just picking her up into a huge hug and saying you silly old thing I do love you you know and blowing a raspberry on her cheek or something would kick her out of it (although sometimes she just kicked me!)
DD1 is now rising 6. Although most of the time she is a fabulous, fun, responsible little girl, she is still very emotionally volatile and still liable to have terrible temper fits. Interestingly these days during a temper tantrum she will often storm off to her bedroom and slam the door, then cuddle up with her comfort blanket and teddy until she calms down, so that tactic obviously sunk in and works for her.
DD2 never had tantrums. DS (now 20 months) is exactly like DD1, but I am so much better at dealing with it this time.