  Let X be a minimal complex surface of general type such that its image via
the canonical map is a surface; we denote by d the degree of the canonical map.
  In this expository work, first of all we recall the known possibilities for
the canonical image and for d when the canonical map is not birational, which
are quite a few, and then we consider the question of producing concrete
examples for all of them.
  We present the two main methods of construction of such examples and we give
several instances of their application. We end the paper outlining the state of
the art on this topic and raising several questions.
