van Enter A.C.D., Shlosman S.B. Provable forst-order transitions for liquid crystal and lattice gauge models with continuous symmetries (29K, LATeX) ABSTRACT. We consider various sufficiently nonlinear sigma models for nematic ordering of RP^{N-1} type and of lattice gauge type with continous symmetries. We rigorously show that they exhibit a first-order transition in the temperature. The result holds in dimension 2 or more for the RP{N-1} models and in dimension 3 or more for the lattice gauge models. In the two-dimensional case our results clarify and solve a recent controversy about the possibilty of such transitions. For lattice gauge models our methods provide the first prof of a first-order transition in a model with a continous gauge symmetry.