  This paper begins with a summary of a powerful formalism for the study of
electronic states in condensed matter physics called "Gauge Theory of
States/Phases of Matter." The chiral anomaly, which plays quite a prominent
role in that formalism, is recalled. I then sketch an application of the chiral
anomaly in 1+1 dimensions to quantum wires. Subsequently, some elements of the
quantum Hall effect in two-dimensional (2D) gapped ("incompressible") electron
liquids are reviewed. In particular, I discuss the role of anomalous chiral
edge currents and of anomaly inflow in 2D gapped electron liquids with
explicitly or spontaneously broken time reversal, i.e., in Hall- and Chern
insulators. The topological Chern-Simons action yielding the transport
equations valid in the bulk of such systems and the associated anomalous edge
action are derived. The results of a general classification of "abelian" Hall
insulators are outlined. After some remarks on induced Chern-Simons actions, I
sketch results on certain 2D chiral photonic wave guides. I then continue with
an analysis of chiral edge spin-currents and the bulk response equations in
time-reversal invariant 2D topological insulators of electron gases with
spin-orbit interactions. The "chiral magnetic effect" in 3D systems and
axion-electrodynamics are reviewed next. This prepares the ground for an
outline of a general theory of 3D topological insulators, including "axionic
insulators". Some remarks on Weyl semi-metals, which exhibit the chiral
magnetic effect, and on Mott transitions in 3D systems with dynamical
axion-like degrees of freedom conclude this review.}
