We consider the question of finding an approximate period in a given string S of length n. Let S′ be a periodic string closest to S under some distance metric. We consider this distance the error of the periodic string, and seek the smallest period that generates a string with this distance to S. In this paper we consider the Hamming and swap distance metrics. In particular, if S is the given string, and S′ is the closest periodic string to S under the Hamming distance, and if that distance is k, we develop an O(nkloglogn) algorithm that constructs the smallest period that defines such a periodic string S′. We call that string the approximate period of S under the Hamming distance. We further develop an O(n 2) algorithm that constructs the approximate period under the swap distance. Finally, we show an O(nlogn) algorithm for finite alphabets, and O(nlog3 n) algorithm for infinite alphabets, that approximates the number of mismatches in the approximate period of the string.