  These lectures give a pedagogical introduction to real and virtual Compton
scattering at low energies.
  We will first discuss real Compton scattering off a point particle as well as
a composite system in the framework of nonrelativistic quantum mechanics.
  The concept of electromagnetic polarizabilities is introduced.
  We then address a description of the Compton-scattering tensor within quantum
field theory with particular emphasis on the derivation of low-energy theorems.
  The importance of a consistent treatment of hadron structure in the use of
electromagnetic vertices is stressed.
  Finally, the reader is introduced to the notion of generalized
polarizabilities in the rapidly expanding field of virtual Compton scattering.
