  Duality refers to two equivalent descriptions of the same theory from
different points of view. Recently there has been tremendous progress in
formulating and understanding possible dualities of quantum many body theories
in $2+1$-spacetime dimensions. Of particular interest are dualities that
describe conformally invariant quantum field theories in $(2+1)d$. These arise
as descriptions of quantum critical points in condensed matter physics. The
appreciation of the possible dual descriptions of such theories has greatly
enhanced our understanding of some challenging questions about such quantum
critical points. Perhaps surprisingly the same dualities also underlie recent
progress in our understanding of other problems such as the half-filled Landau
level and correlated surface states of topological insulators. Here we provide
a pedagogical review of these recent developments from a point of view geared
toward condensed matter physics.
