  This is an introductory review to localization techniques in supersymmetric
two-dimensional gauge theories. In particular we describe how to construct
Lagrangians of N=(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 N=(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.
