  Given a projective variety X and a smooth projective curve C one may consider
the moduli space of maps C --> X. This space admits certain compactification
whose points are called quasi-maps. In the last decade it has been discovered
that in the case when X is a (partial) flag variety of a semi-simple algebraic
group G (or, more generally, of any symmetrizable Kac-Moody Lie algebra) these
compactifications play an important role in such fields as geometric
representation theory, geometric Langlands correspondence, geometry and
topology of moduli spaces of G-bundles on algebraic surfaces, 4-dimensional
super-symmetric gauge theory (and probably many others). This paper is a survey
of the recent results about quasi-maps as well as their applications in
different branches of representation theory and algebraic geometry.
