The transcription factor p53 plays a central role in maintaining genomic integrity. Recent experiments in MCF7 cells have shown that p53 protein level rises and falls in distinct pulses in response to DNA damage. The amplitudes of and intervals between pulses seem to be independent of the extent of damage, and some cells generate regular pulses of p53 over many days. Identifying the molecular mechanisms responsible for such interesting behavior is an important and challenging problem. This paper describes four dual-feedback mechanisms that combine both positive and negative feedback loops, which have been identified in the signaling network responsible for p53 regulation. Mathematical models of all four mechanisms are analyzed to determine if they are consistent with experimental observations and to characterize subtle differences among the possible mechanisms. In addition, a novel molecular mechanism is proposed whereby p53 pulses may induce, at first, cell cycle arrest and, if sustained, cell death. The proposal accounts for basic features of p53-mediated responses to DNA damage and suggests new experiments to probe the dynamics of p53 signaling.