  We discuss the the notion of a partial dynamical symmetry (PDS), for which a
prescribed symmetry is obeyed by only a subset of solvable eigenstates, while
other eigenstates are strongly mixed. We present an explicit construction of
Hamiltonians with this property, including higher-order terms, and portray
their significance for spectroscopy and shape-phase transitions in nuclei. The
occurrence of both a single PDS, relevant to stable structures, and of several
PDSs, relevant to coexistence phenomena, are considered.
