my son is 15 months old. He has just started this...almost like "cool, look what I can do" then he kicks and screams.
At home...I ignore him. I ask him if he's done when he stops throwing the fit...and then get back to what we were doing before hand. If it's too out of hand (and no he hasn'd done anything too out of hand yet) I will put him in his room, on the floor...so that he won't hurt himself and then walk away. (I don't put him in his crib because I don't want him thinking his bed is a place of punishment)
Most kids throw fits because it get's a reaction from the parent/grandparent/adult. Also, at 15 months, they can't tell you...hey...my diaper is chaffing or I'm thirsty or can I have a banana. they get frustrated.
Oh...and in public...as long as he's not hurting someone...I stand there un reactive...until he stops. My oldest son used to do this when we went to the grocery. Throw himself RIGHT DOWN on the floor. I'd turn around and wait...never giving him the reaction he wanted. It stopped. (Now at church, I remove would remove him to another room and let him do his thing. Time and place...time and place)
I also repeat the mantra...when things get especially crazy...
This too shall pass. This too shall Pass. This too shall pass.
Smiles to you. Good luck.
Fun stage, huh?