Harmonic analysis on reductive groups

Fourier theory tells us how to express a function as a superposition (sum or integral) of basic waves (sines and cosines). These results fit into the following general paradigm: a conjugation invariant function on a group G can be expressed as a superposition of characters of irreducible representations. I will survey various realizations of this paradigm, starting with historical results and classical examples, and ending with more recent results and areas of ongoing research.