  The irreducible decomposition of a unitary representation often contains
continuous spectrum when restricted to a non-compact subgroup. The author
singles out a nice class of branching problems where each irreducible summand
occurs discretely with finite multiplicity (admissible restrictions). Basic
theory and new perspectives of admissible restrictions are presented from both
analytic and algebraic view points. We also discuss some applications of
admissible restrictions to modular varieties and $L^p$-harmonic analysis.
