  We wish to understand how irreducible representations of a group G behave
when restricted to a subgroup G' (the branching problem). Our primary concern
is with representations of reductive Lie groups, which involve both algebraic
and analytic approaches. We divide branching problems into three stages:
  (A) abstract features of the restriction;
  (B) branching laws (irreducible decompositions of the restriction); and
  (C) construction of symmetry breaking operators on geometric models.
  We could expect a simple and detailed study of branching problems in Stages B
and C in the settings that are {\it{a priori}} known to be "nice" in Stage A,
and conversely, new results and methods in Stage C that might open another
fruitful direction of branching problems including Stage A.
  The aim of this article is to give new perspectives on the subjects, to
explain the methods based on some recent progress, and to raise some
conjectures and open questions.
