Peristaltic transport of an incompressible viscous fluid in an asymmetric channel is studied under long-wavelength and low-Reynolds number assumptions. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The flow is investigated in a wave frame of reference moving with velocity of the wave. The effects of phase difference, varying channel width and wave amplitudes on the pumping characteristics, streamline pattern, trapping, and reflux phenomena are investigated. The limits on the time averaged flux for trapping and reflux are obtained. It is observed that the pumping against pressure rise, trapping and reflux layer exists only when cross-section of the channel varies. The peristaltic waves on the walls with same amplitude propagating in phase produce zero flux rate as the channel cross-section remains same through out. The trapping and reflux regions increase as the channel becomes more and more symmetric and the maximum occurs for the symmetric channel.