Qualitative simulation is a key inference process in qualitative causal reasoning. However, the precise meaning of the different proposals and their relation with differential equations is often unclear. In this paper, we present a precise definition of qualitative structure and behavior descriptions as abstractions of differential equations and continuously differentiable functions. We present a new algorithm for qualitative simulation that generalizes the best features of existing algorithms, and allows direct comparisons among alternate approaches. Starting with a set of constraints abstracted from a differential equation, we prove that the QSIM algorithm is guaranteed to produce a qualitative behavior corresponding to any solution to the original equation. We also show that any qualitative simulation algorithm will sometimes produce spurious qualitative behaviors: ones which do not correspond to any mechanism satisfying the given constraints. These observations suggest specific types of care that must be taken in designing applications of qualitative causal reasoning systems, and in constructing and validating a knowledge base of mechanism descriptions.