Simple chemical reactions in low dimensions under a great variety of conditions lead to the spontaneous formation of spatial patterns and to associated "anomalous" (usually slower) rate laws for the global densities of the reacting species. The kinetic anomalies arise as a consequence of the spatial distribution of the reactants. The standard exponents reflect a random spatially homogeneous distribution; the law of mass action presumes that such a distribution is maintained at all times, even as the reaction proceeds. This, in turn, presumes an efficient mixing mechanism. What might such a mixing mechanism be? Examples include physical stirring, convection, and diffusion. Deviations from a homogeneous distribution arise when there is no mixing or when there is insufficient mixing. Diffusion is not a particularly effective mixing mechanism, especially in low dimensions. As a result, initial spatial density fluctuations lead to spatial ordering at later times, and this spatial ordering in turn is manifested through nonclassical rate laws. In this lecture we discuss the underlying ideas that lead to this behavior, as well as the different physical manifestations of spatial ordering. We present analytical and numerical methods that have been used to address these problems. We also briefly mention experimental evidence (not plentiful) of these effects, and some completely different contexts (unrelated to chemical reactions) where these same descriptions and effects are applicable.