Streamline patterns and their local and global bifurcations in a two-dimensional planar and axisymmetric peristaltic flow for an incompressible Newtonian fluid have been investigated. An analytical solution for the stream-function is found under a long-wavelength and low-Reynolds number approximation. The problem is solved in a moving coordinate system where a system of nonlinear autonomous differential equations can be established for the particle paths. Local bifurcations and their topological changes are inspected using methods of dynamical systems. Three different flow situations manifest themselves: backward flow, trapping or augmented flow. The transition between backward flow to trapping corresponds to a bifurcation of co-dimension one, in which a non-simple degenerate point changes its stability to form heteroclinic connections between saddle points that enclose recirculating eddies. The transition from trapping to augmented flow is a bifurcation of co-dimension two, in which heteroclinic saddle connections of adjacent waves coalesce below wave troughs. The coalescing of saddle nodes on the longitudinal axis produces a degenerate point with six heteroclinic connections (degenerate saddle). As the parameter is increased, the degenerate saddle bifurcates to saddles nodes which lift off the centerline. These bifurcations are summarized in a global bifurcation diagram. Theoretical results are compared with the experimental data. Published by Elsevier B.V.