Let k, t and m be positive integers. A k-multiset of [m] is a collection of k integers from the set {1,...,m} in which the integers can appear more than once. We use graph homomorphisms and existing theorems for intersecting and t-intersecting k-set systems to prove new results for intersecting and t-intersecting families of k-multisets. These results include a multiset version of the Hilton-Milner theorem and a theorem giving the size and structure of the largest t-intersecting family of k-multisets of an m-set when m ≤ 2k-t.