We consider the foundations of charged-particle acceleration at collisionless shocks in which the average magnetic field is perpendicular to the shock-normal direction. Our shock model, which is a numerical integration of the trajectories of an ensemble of test particles in specified shock fields, includes an isotropic and homogeneous, Kolmogorov-like, fluctuating magnetic field superimposed on a background field. The field is fully three-dimensional so that transverse diffusion is possible. Moreover, the scattering is a natural consequence of the particle interaction with the fields and is not treated in an ad hoc manner. We consider two distinctly different regimes for the fluctuating component of the field: (1) one in which the field line mixing (or field line random walk), due to large-scale fluctuations in the random component of the field, dominates the perpendicular diffusion coefficient and (2) one in which the the large-scale variations in the field are removed and the perpendicular diffusion coefficient is smaller. In the first case, the transverse diffusion coefficient is larger than in the second case owing to the field line mixing which leads to field line loops which intersect the shock in several places and particles can become temporarily trapped as the field line convects through the shock. In the latter case, the loops are on a spatial scale that is shorter than the particle gyroradii, and the only means for particles to remain near the shock is by ''resonant'' transverse diffusion. Comparisons between previous numerical models and the diffusive theory will be discussed as well as the importance of these ideas to astrophysical applications such as particle injection to anomalous cosmic ray energies at the solar wind termination shock. The results from these simulations suggest that simple scattering off of convected magnetic fluctuations cannot readily account for the acceleration of nonaccelerated pickup ions. ¿ American Geophysical Union 1996