Confinement in a pure gauge theory at non-zero temperature may be discussed in terms of an order parameter which transforms under a global symmetry group, the center of the gauge group. Integrating out all degrees of freedom except this order parameter generates an effective scalar field theory for the order parameter, globally invariant under the center symmetry. We argue that the effective theory possesses only short-range couplings, and hence that the finite-temperature confinement phase transition (when continuous) is accompanied by long-range fluctuations only in the order parameter. Universality ideas then lead to predictions for the critical properties of U(1), Z(N), and SU(N) gauge theories for all dimensionalities of space-time. An explicit renormalization-group calculation is presented for the U(1) gauge theory in (2 + 1) dimensions, the results of which fit the general picture.