  We say a representation V of a group G has stability if its multiplicities
m^{G}_{V}(\lambda) is dependent only on some equivalence class of \lambda for a
sufficiently large parameter \lambda. In this paper, we prove that the
restriction of a holomorphic discrete series representation with respect to any
holomorphic symmetric pairs has stability. As a corollary, we give a necessary
and sufficient condition on multiplicity-freeness of the branching laws in this
setting. This condition is same as the sufficient condition given by the theory
of visible actions. We prove a general theorem before we show the stability of
holomorphic discrete series representations. Using the general theorem, we also
show the stability on quasi-affine spherical homogeneous spaces and the
stability of K-type of unitary highest weight modules. We also show that two
branching laws of a holomorphic discrete series representation coincide if two
subgroups are in same \epsilon-family.
