Can you just replace cleats instead of buying a whole new set of pedals/cleats?

I realized why my left food keeps popping out of my pedal, b/c the latchy part of the cleat (Look style) is only half there, the other half must have snapped off from walking on it or something.

I got the pedals a few years ago...do the new cleats still fit the older pedals??

Thanks!

K.