In the late 1950s, an important problem in number theory was to extend the notion of $L$-functions attached to cuspforms on the upper-half plane to automorphic forms on reductive groups. Langlands’s work on non-abelian Harmonic analysis, namely the problem of the spectral decomposition of automorphic forms, led him to a general notion of $L$-functions attached to cuspforms. We give an introduction to the spectral decomposition of automorphic forms and discuss some contemporary problems.