  The methods of quantum field theory are widely used in condensed matter
physics. In particular, the concept of an effective action was proven useful
when studying low temperature and long distance behavior of condensed matter
systems. Often the degrees of freedom which appear due to spontaneous symmetry
breaking or an emergent gauge symmetry, have non-trivial topology. In those
cases, the terms in the effective action describing low energy degrees of
freedom can be metric independent (topological). We consider a few examples of
topological terms of different types and discuss some of their consequences. We
will also discuss the origin of these terms and calculate effective actions for
several fermionic models. In this approach, topological terms appear as phases
of fermionic determinants and represent quantum anomalies of fermionic models.
In addition to the wide use of topological terms in high energy physics, they
appeared to be useful in studies of charge and spin density waves, Quantum Hall
Effect, spin chains, frustrated magnets, topological insulators and
superconductors, and some models of high-temperature superconductivity.
