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We wish to understand how irreducible representations of a group G behavewhen restricted to a subgroup G' (the branching problem). Our primary concernis with representations of reductive Lie groups, which involve both algebraicand 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 Band 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 anotherfruitful direction of branching problems including Stage A.  The aim of this article is to give new perspectives on the subjects, toexplain the methods based on some recent progress, and to raise someconjectures and open questions.

A program for branching problems in the representation theory of real reductive groups