Abstract: Let G be a group and H a subgroup. Frobenius showed in 1898 how to "enlarge" a representation of H to a representation of G. His method, now called induction, rapidly became a useful technical tool in algebra and harmonic analysis and was adapted by others in various ways. For instance, in 1965 Bott made a systematic study of induction methods based on invariant elliptic differential operators in the context of compact Lie groups, which led to generalizations of the Weyl character formula. I will review and update Bott's work and discuss some applications to K-theory.