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It has recently been recognized that significant numbers of medium-mass back holes (of order 10 solar masses) should form in globular clusters during the early stages of their evolution. Here we explore the dynamical and observational consequences of the presence of such a primordial black-hole population in a globular cluster. The holes initially segregate to the cluster cores, where they form binary and multiple black-hole systems. The subsequent dynamical evolution of the black-hole population ejects most of the holes on a relatively short timescale: a typical cluster will retain between zero and four black holes in its core, and possibly a few black holes in its halo. The presence of binary, triple, and quadruple black-hole systems in cluster cores will disrupt main-sequence and giant stellar binaries; this may account for the observed anomalies in the distribution of binaries in globular clusters. Furthermore, tidal interactions between a multiple black-hole system and a red giant star can remove much of the red giant's stellar envelope, which may explain the puzzling absence of larger red giants in the cores of some very dense clusters.