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Abstract: A multi-latin square of order and index is an array of multisets, each of cardinality , such that each symbol from a fixed set of size occurs times in each row and times in each column. A multi-latin square of index is also referred to as a -latin square. A -latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in -latin square of order embeds in a -latin square of order , for each , thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable -latin squares of order for each . We also show that for each , there exists some finite value such that for all , every -latin square of order is separable. We discuss the connection between -latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and -latin trades. We also enumerate and classify -latin squares of small orders. [Copyright &y& Elsevier]