In real Euclidean space a reflection is an orthogonal transformation which fixes every vector of some hyperplane. Finite groups generated by reflections in a real vector space have been studied in great depth and they play a central role in many branches of mathematics, particularly in the theory of Lie groups and Lie algebras. In these lectures we shall study an important generalisation: reflections in complex space and their polynomial invariants. Particular attention will be given to examples in low dimensions and their connections with quaternions and octonions. If time permits, the new invariant theoretic proof, due to Lehrer, of Steinberg's theorem will be given: this shows that the pointwise stabiliser in a reflection group of a set of vectors is itself a reflection group.