It sounds like the terrible 2's. My youngest started with it early.  She would throw herself on the floor and just scream.

So I started to count to 5 and if she wasn't up then I'd pick her up and put her in a safe environment to cry it out, her crib in her room. I told her I'd be back when she stopped which probably took no longer than 5 minutes. After I go back to get her and I'd chat w/her and tell her "no more throwing yourself on the floor", tell me what you want or show me what you want.  Needless to say she learned to count to 5 at an early age ;).

If you're in public and she does that, leave the situation, I have left plenty of baskets full at the store.

HI Lyndseyb
At 13 months old, a baby may do this for many reasons. usually its for attention, other times it could be frustration because she doesn’t know how to express her emotions in another way, another reason could be she not getting her own way.
But at 13 months old she is still a baby, and the tantrums may stop.
For now, I would try ignoring her, don't even look at her when she has a tantrum, just make sure she out of harms way.
Or simply say "that’s not going to work anymore" (that’s my favorite one for friend children)
and turn around.
When she learns she's not getting your goat (so to speak), and she stops, turn back around, don’t say any thing about the tantrum, then give she some attention, and if she starts again, turn back around, it may take a few times before she learns, but be careful not to teach her that after having a tantrum she'll get goodies, reward her with love and kindness.

I do agree that miss-distraction by using a toy does work, as well as soft music, finding some thing to look at out the window, like a bus, birds, or some thing she may not have seen before.
All of theses strategies have worked, especially for my children, and family and friends children’s.
Be consistent with your method which ever rout you choose
I hope you can find one that works for you and your baby*
Good Luck*

Our daughter is 2 1/2 and I firmly believe it is the terrible ones not twos. She started tantrums just after her first birthday.  We did one of two things.  If we could see it coming we would warn her to stop or give her time out.  It sounds young to do time out but it was the only thing we could think of that we hadn't tried and it worked a charm.  We would leave her in her room for a minute, she would yell and bang the door but when we went back in there she would stop. 

If we didn't see it coming and it started, even in public we would just leave her alone.  At times this was extrememly embarressing but I quickly learned that those starting and judging usually didn't have kids of their own so what did they know. 

A friend of ours had a son who would get on all fours and bang his head on the floor when he was throwing a tantrum. For a while they would pick him up and keep him from doing so, but this was only a temporary solution to the problem. Once they put him down if he didn't get what he wanted he was right back to banging his head.

After one week of ignoring him, letting him sit there and bang his head on the ground, and he was done with it. He learned he could no longer get what he wanted that way.

Your child understands pain - they won't intentionally hurt themselves and in a tantrum what is the worse they can do? In my friend's case, he could knock himself out (which he did once). So, he wakes up after a few seconds realized he had knocked himself out, his head hurt really bad, and he still didn't get the ice cream he wanted. He understood, "maybe I shouldn't do this anymore, it's stupid and it hurts."

Sometimes they have to teach themselves their own lessons.