  Let S be a cyclic n-ary sequence. We say that S is a {\it universal cycle}
((n,k)-Ucycle) for k-subsets of [n] if every such subset appears exactly once
contiguously in S, and is a Ucycle packing if every such subset appears at most
once. Few examples of Ucycles are known to exist, so the relaxation to packings
merits investigation. A family {S_n} of (n,k)-Ucycle packings for fixed k is a
near-Ucycle if the length of S_n is $(1-o(1))\binom{n}{k}$. In this paper we
prove that near-(n,k)-Ucycles exist for all k.
