Undergraduate Research in Mathematics

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Dougggggg
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I plan on one day going to graduate school for Math or Physics, depending on what graduate programs I get into and where my interest finally ends up falling. I am already doing some undergraduate research in physics and I began to wonder about if REU's in math were common and if so, what level do you typically have to be at in order to get involved in them?
 
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Sure they are. The question I would have for you is, how many math classes have you had so far? You could honestly do research in anything math-related, but obviously the higher up you are, the more options you will have as far as choosing an interesting topic. I just looked online and found someone that did a project on continued fractions:

http://archives.math.utk.edu/articles/atuyl/confrac/

As far as I know, you really don't need that much background to do this and it seems like an interesting topic. However, if you want to do something on linear algebra, modern algebra, number theory, etc.. you might want to get some sophomore/junior classes knocked out and you'll have a better understanding of what to do.