The structure of low Mach number parallel and quasi-parallel shocks is investigated by using a one-dimensional electromagnetic hybrid (fluid electrons, particle ions) code. It is shown that both structure of and ion dissipation at these shocks can be divided into two distinct regimes. In the parallel limit, the shock consists of large amplitude electromagnetic waves on the magnetosonic-whistler branch. The amount of ion heating and scattering by these waves is shown to be minimal, and as a result the shock does not decouple from the piston. Instead, the required hotter downstream ion distribution function is formed by the supersposition of two distinct populations of incident and piston reflected ions. This distribution function is shown to be unstable near the shock transition region, maintaining the magnetosonic-whistler waves, but is stable further downstream. The structure of quasi-parallel shocks is found to be considerably different. Initially, the shock consists of phase standing dispersive whistlers in the upstream, whose last cycle constitutes the shock ramp. These waves make little or no contribution to ion heating. At later times, a longer wavelength whistler wave packet is formed upstream of the shock. These waves are shown to be excited by the backstream (reflected or leaked) ions via the electromagnetic ion beam instability. Unlike the phase standing whistlers, these waves result in considerable local deceleration and heating of the solar wind plasma. As a result, the transition from upstream to downstream plasma is not monotonic but rather, it occurs in an intermittent fashion. Nevertheless, unlike high Mach number quasi-parallel shocks that have been shown to periodically reform, low Mach number shocks are found to be steady. ¿ American Geophysical Union 1990