A mechanism for the acceleration of pickup ions by repeated reflection from the electrostatic cross-shock potential of a quasi-perpendicular shock was proposed independently by Zank et al. [1996b> and Lee et al. [1996>. The acceleration mechanism, known variously as Multiply Reflected Ion (MRI) acceleration or shock surfing, was studied by these authors in the limit of an idealized shock, which possessed neither fine-scale structure (distinct foot, ramp, or overshoot) nor pickup ion scattering turbulence. Here the acceleration of pickup ions at cometary and interplanetary shocks and at the termination shock is studied in the test particle limit on the basis of a particle-mesh simulation. All simulations assume a shell distribution for the pickup ions (either pickup protons or helium), and the dynamics of pickup ions propagating in a fixed electromagnetic field profile are investigated. The effect of a shock foot, ramp and overshoot on the acceleration of pickup ions at perpendicular and oblique shocks is described. The acceleration of pickup ions in the presence of strong turbulence inside the foreshock region is also addressed. For quasi-perpendicular shocks with structure, we obtained the following results. First, the accelerated H+ and He+ spectrum is a very hard power law ∝E-k, k=0.92--1.2, which is much harder than that predicted by diffusive shock acceleration. Also, as &thgr;bn, the angle between the upstream magnetic field and shock normal, decreases from 90¿, the accelerated pickup ion spectrum flattens until MRI acceleration ceases. Second, the fine structure of the shock is found to reduce slightly the maximum energy gain for an accelerated pickup ion compared to that gained at an unstructured shock. Third, a flat turbulence spectrum is found to lead to an increase in the maximum energy for transmitted pickup ions, whereas a power law turbulence spectrum leads to a modest reduction in the maximum pickup ion energy gain. However, the basic MRI acceleration mechanism continues to operate in the presence of turbulent magnetic fluctuations and the characteristic hard spectra are preserved. Fourth, MRI acceleration is found to work only for those shocks for which &thgr;bn≥60¿--70¿. The results from the test particle simulations described here are also used to interpret particle acceleration in self-consistent hybrid simulations. ¿ 1998 American Geophysical Union