Starting with an analysis of chiral edge currents in 2D electron gases exhibiting the quantum Hall effect I will discuss the role of anomalous chiral edge currents and of anomaly inflow in 2D insulators with explicitly broken parity and time-reversal and in time-reversal-invariant 2D topological insulators exhibiting edge spin-currents. I will also briefly discuss the topological field theories that yield the correct response equations for the 2D bulk of such materials. After a short excursion into the theory of 3D topological insulators I will then discuss an analogue of the QHE in 4 dimensions. Among other things, this will lead us to a model of dark matter and dark energy involving an axion that couples to the Pontryagin density of a massive gauge field.