The main result of the author is a Liouville type theorem for harmonic maps with domain \(M\), a complete Riemannian manifold of nonnegative Ricci curvature, and range \(N\), a simply-connected complete Riemannian manifold with sectional curvature bounded above by \(-a^2\), \(a>0\). It states that a harmonic map \(f: M\to N\) whose image is contained in a horoball \(B_c\) centered at \(c(+\infty)\) with respect to a unit speed geodesic \(c\), is necessarily a constant. A vanishing theorem for harmonic maps is proved in the same setting, with the curvature constraint on the range relaxed to nonpositivity. These results are used to derive nonexistence of complete metrics with preassigned Ricci curvature for domains or horoballs.