A.C.D. van Enter, S.B.Shlosman First-order transitions for n-vector models in two and more dimensions; rigorous proof (20K, latex) ABSTRACT. We prove that various SO(n)-invariant n-vector models with interactions which have a deep and narrow enough minimum have a first-order transition in the temperature. The result holds in dimension two or more and is independent on the natur of the low-temperature phas.