  We consider a variety of nearest-neighbor spin models defined on the
d-dimensional hypercubic lattice Z^d. Our essential assumption is that these
models satisfy the condition of reflection positivity. We prove that whenever
the associated mean-field theory predicts a discontinuous transition, the
actual model also undergoes a discontinuous transition (which occurs near the
mean-field transition temperature), provided the dimension is sufficiently
large or the first-order transition in the mean-field model is sufficiently
strong. As an application of our general theory, we show that for d
sufficiently large, the 3-state Potts ferromagnet on Z^d undergoes a
first-order phase transition as the temperature varies. Similar results are
established for all q-state Potts models with q>=3, the r-component cubic
models with r>=4 and the O(N)-nematic liquid-crystal models with N>=3.
