  We establish large deviation principles (LDPs) for empirical measures
associated with a sequence of Gibbs distributions on $n$-particle
configurations, each of which is defined in terms of an inverse temperature $%
\beta_n$ and an energy functional consisting of a (possibly singular)
interaction potential and a (possibly weakly) confining potential. Under fairly
general assumptions on the potentials, we use a common framework to establish
LDPs both with speeds $\beta_n/n \rightarrow \infty$, in which case the rate
function is expressed in terms of a functional involving the potentials, and
with speed $\beta_n =n$, when the rate function contains an additional entropic
term. Such LDPs are motivated by questions arising in random matrix theory,
sampling, simulated annealing and asymptotic convex geometry. Our approach,
which uses the weak convergence method developed by Dupuis and Ellis,
establishes LDPs with respect to stronger Wasserstein-type topologies. Our
results address several interesting examples not covered by previous works,
including the case of a weakly confining potential, which allows for rate
functions with minimizers that do not have compact support, thus resolving
several open questions raised in a work of Chafa\"{\i} et al.
