Gauge anomalies arise in the presence of massless chiral fermions and signal the absence of classical chiral symmetry in the quantum system. Historically chiral anomalies were first observed in perturbative calculations of certain 1-loop scattering amplitudes. Today there is a beautiful non-perturbative description of chiral symmetry breaking, using methods from index theory and global analysis. The aim of this talk is to give some background to anomalies and explain how some of the ideas from physics lead to interesting constructions in mathematics.