  A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of
$\binom{n+t-1}{t}$ integers from [n] with the property that each t-multiset of
[n] appears exactly once consecutively in the sequence. For such a sequence to
exist it is necessary that n divides $\binom{n+t-1}{t}$, and it is reasonable
to conjecture that this condition is sufficient for large enough n in terms of
t. We prove the conjecture completely for t in {2,3} and partially for t in
{4,6}. These results also support a positive answer to a question of Knuth.
