Let N be a connected, simply connected Lie group with Lie algebra \({\mathfrak n}\), and let \({\mathfrak u}({\mathfrak n})\) be the universal enveloping algebra of \({\mathfrak n}\). Then elements of \({\mathfrak u}({\mathfrak n})\) act as (left invariant) differential operators on N. Now suppose that N has a discrete, cocompact subgroup \(\Gamma\), and let \(\rho\) denote the unitary representation of N on \(L^ 2(\Gamma \backslash N)\) coming from right translation; \(\rho\) is a discrete direct sum of irreducibles. Let [N:\(\Gamma]\) \^ denote the set of representations appearing in \(\rho\). The authors show that \(L\in {\mathfrak u}({\mathfrak n})\) is locally solvable on N if \(\pi\) (L) has a bounded right inverse \(A_{\pi}\) for ”almost all” \(\pi\in [N:\Gamma]\) \^, and if \(\| A_{\pi}\|\) satisfies some growth conditions as a function of \(\pi\). They also give a version of the above result which yields semiglobal solvability for certain L. (This version can be combined with a theorem in [the first author and L. P. Rothschild [”Solvability of transversally elliptic differential operators on nilpotent Lie groups”, Am. J. Math. (to appear)] to yield a fundamental solution for appropriate L.)