The pioneering work of Langlands has established the theory of reductive algebraic groups and their representations as a key part of modern number theory. I will survey classical and modern results in the representation theory of reductive groups over local fields (the fields of real, complex, or p-adic numbers, or of Laurent series over finite fields) and discuss how they relate to Langlands' ideas, as well as to the various reflections of the basic mathematical idea of symmetry in arithmetic and geometry.