Supersymmetric localization in two dimensions

This is an introductory review to localization techniques in supersymmetric two-dimensional gauge theories. In particular, we describe how to construct Lagrangians of $ \newcommand{\re}{{\rm Re}~} \newcommand{\mat}[1]{\left(\begin{array}{cc}#1\end{array}\right)} \newcommand{\cN}{\mathcal{N}} \cN = (2, 2)$ theories on curved spaces, and how to compute their partition functions and certain correlators on the sphere, the hemisphere and other curved backgrounds. We also describe how to evaluate the partition function of $ \newcommand{\re}{{\rm Re}~} \newcommand{\mat}[1]{\left(\begin{array}{cc}#1\end{array}\right)} \newcommand{\cN}{\mathcal{N}} \cN = (0, 2)$ theories on the torus, known as the elliptic genus. Finally we summarize some of the applications, in particular to probe mirror symmetry and other non-perturbative dualities.