Diffusive shock acceleration (DSA), believed to be responsible for the non-thermal, high-energy particles found in diverse astronomical systems, is not yet fully understood. For non-relativistic (NR) shocks, DSA yields a flat, p~2 spectral index, in accordance with observations, if the particle distribution function (PDF) is assumed isotropic. We correct previous analyses to explain why the spectrum does not change in the presence of PDF anisotropies. New insights are obtained by generalizing DSA theory to N-dimensions, relativistic shocks, and arbitrary particle diffusion functions (DF). The flat, p~2 spectrum is shown to be independent of N for NR shocks. The spectrum is independent of the DF only in 1D, where it curiously asymptotes to p~2 in both NR and ultra-relativistic shock limits. We generalize to 2D expressions for the relation between p and the PDF anisotropy, and for p in an arbitrary shock with an isotropic DF. Here, eigenfunctions of the transport equation reduce to elliptic cosine functions, providing a rigorous solution to the problem. Unlike previous claims, we show that DSA in 3D can produce very hard spectra, up to p~1, similar to the spectra inferred in pulsar wind nebulae (PWNe). However, such spectra require strong confinement to the shock front, ruling out DSA as the origin of PWN synchrotron emission.