In this review we discuss the recently described roaming mechanism for chemical reactions from the point of view of nonlinear dynamical systems in phase space. The recognition of the roaming phenomenon shows the need for further developments in our fundamental understanding of basic reaction dynamics, as is made clear by considering some questions that cut across most studies of roaming: Is the dynamics statistical? Can transition state theory be applied to estimate roaming reaction rates? What role do saddle points on the potential energy surface play in explaining the behavior of roaming trajectories? How do we construct a dividing surface that is appropriate for describing the transformation from reactants to products for roaming trajectories? How should we define the roaming region? We show that the phase space perspective on reaction dynamics provides the setting in which these questions can be properly framed and answered. We illustrate these ideas by considering photodissociation of formaldehyde. The phase-space formulation allows an unambiguous description of all possible reactive events, which also allows us to uncover the phase space mechanism that explains which trajectories roam, as opposed to evolving toward a different reactive event.