  The complex analytic methods have found a wide range of applications in the
study of multiplicity-free representations. This article discusses, in
particular, its applications to the question of restricting highest weight
modules with respect to reductive symmetric pairs. We present a number of
multiplicity-free branching theorems that include the multiplicity-free
property of some of known results such as the Clebsh--Gordan--Pieri formula for
tensor products, the Plancherel theorem for Hermitian symmetric spaces (also
for line bundle cases), the Hua--Kostant--Schmid $K$-type formula, and the
canonical representations in the sense of Vershik--Gelfand--Graev. Our method
works in a uniform manner for both finite and infinite dimensional cases, for
both discrete and continuous spectra, and for both classical and exceptional
cases.
