  A number of examples have demonstrated the failure of the
Landau-Ginzburg-Wilson(LGW) paradigm in describing the competing phases and
phase transitions of two dimensional quantum magnets. In this paper we argue
that such magnets possess field theoretic descriptions in terms of their slow
fluctuating orders provided certain topological terms are included in the
action. These topological terms may thus be viewed as what goes wrong within
the conventional LGW thinking. The field theoretic descriptions we develop are
possible alternates to the popular gauge theories of such non-LGW behavior.
Examples that are studied include weakly coupled quasi-one dimensional spin
chains, deconfined critical points in fully two dimensional magnets, and two
component massless $QED_3$. A prominent role is played by an anisotropic O(4)
non-linear sigma model in three space-time dimensions with a topological theta
term. Some properties of this model are discussed. We suggest that similar
sigma model descriptions might exist for fermionic algebraic spin liquid
phases.
