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We give a pedagogical introduction to the study of supersymmetric partitionfunctions 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 gaugetheories can flow to non-trivial fixed points in the infrared. In the presenceof 3D $\mathcal{N}{=}2$ supersymmetry, many exact results are known about thestrongly-coupled infrared, due in good part to powerful localizationtechniques. We review some of these techniques and emphasize some more recentdevelopments, which provide a simple and comprehensive formalism for the exactcomputation of half-BPS observables on closed three-manifolds (partitionfunctions and correlation functions of line operators). Along the way, we alsoreview simple examples of 3D infrared dualities. The computation ofsupersymmetric partition functions provides exceedingly precise tests of thesedualities.

Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review