A proper coloring of a Latin square of order n is an assignment of colors to its elements triples such that each row, column and symbol is assigned n distinct colors. Equivalently, a proper coloring of a Latin square is a partition into partial transversals. The chromatic index of a Latin square is the least number of colors needed for a proper coloring. We study the chromatic index of the cyclic Latin square which arises from the addition table for the integers modulo n. We obtain the best possible bounds except for the case when n=2 is odd and divisible by 3. We make some conjectures about the chromatic index, suggesting a generalization of Ryser's conjecture (that every Latin square of odd order contains a transversal).