Suppose $n$ circular arcs of lengths $\len_i \in [0,1],0\leq i<n$, are placed uniformly at random on a unit length circle. We study the maximum overlap, i.e., the number of arcs that overlap at the same position of the circle. In particular, we give almost exact tail bounds for this random variable. By applying these tail bounds we can characterize the expected maximum overlap exactly up to constant factors in lower order terms. We illustrate the strength of our results by presenting new performance guarantees for three algorithmic applications: minimizing rotational delays for disks, scheduling accesses to parallel disks, and allocating memory blocks to limit cache interference misses.