  A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of
multisets, each of cardinality $k$, such that each symbol from a fixed set of
size $n$ occurs $k$ times in each row and $k$ times in each column. A
multi-latin square of index $k$ is also referred to as a $k$-latin square. A
$1$-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 $k$-latin square of order
$m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus
generalizing Evans' Theorem. Exploiting this result, we show that there exist
non-separable $k$-latin squares of order $n$ for each $n\geq k+2$. We also show
that for each $n\geq 1$, there exists some finite value $g(n)$ such that for
all $k\geq g(n)$, every $k$-latin square of order $n$ is separable.
  We discuss the connection between $k$-latin squares and related combinatorial
objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares
and $k$-latin trades. We also enumerate and classify $k$-latin squares of small
orders.
