  We give a pedagogical introduction to the study of supersymmetric partition
functions of 3D $\mathcal{N}{=}2$ supersymmetric Chern-Simons-matter theories
(with an $R$-symmetry) on half-BPS closed three-manifolds---including $S^3$,
$S^2 \times S^1$, and any Seifert three-manifold. Three-dimensional gauge
theories can flow to non-trivial fixed points in the infrared. In the presence
of 3D $\mathcal{N}{=}2$ supersymmetry, many exact results are known about the
strongly-coupled infrared, due in good part to powerful localization
techniques. We review some of these techniques and emphasize some more recent
developments, which provide a simple and comprehensive formalism for the exact
computation of half-BPS observables on closed three-manifolds (partition
functions and correlation functions of line operators). Along the way, we also
review simple examples of 3D infrared dualities. The computation of
supersymmetric partition functions provides exceedingly precise tests of these
dualities.
