In this paper we explore the notion of periods of a string. A period can be thought of as a shift that causes the string to match over itself. We obtain two sets of necessary and sufficient conditions for a set of integers to be the set of periods of some string (what we call the correlation of the string). We show that the number of distinct correlations of length n is independent of the alphabet size and is of order nlogn. By using generating function methods we enumerate the strings having a given correlation, and investigate certain related questions.