Direct detection searches for dark matter have made significant advances in their level of sensitivity and are expected to improve further in the near future. The results of these experiments are usually interpreted in a limited context of certain kinds of interactions between dark matter and standard matter, but the full range of possible interactions is much richer. We present the general non-relativistic effective theory of dark matter (DM) direct detection, which describes this wider range of possibilities. We find several operators that lead to novel nuclear responses. These responses differ significantly from the standard minimal WIMP cases in their relative coupling strengths to various elements, changing how the results from different experiments should be compared against each other. Response functions are evaluated for common nuclei using standard shell model techniques.