Research in Qualitative Reasoning builds and uses discrete symbolic models of the continuous world. Inference methods such as qualitative simulation are grounded in the theory of ordinary differential equations. We argue here that cognitive mapping -- building and using symbolic models of the large-scale spatial environment -- is a highly appropriate domain for qualitative reasoning research. We describe the Spatial Semantic Hierarchy (SSH), set of distinct representations for space, each with its own ontology, each with its own mathematical foundation, and each abstracted from the levels below it. At the control level, the robot and its environment are modeled as a continuous dynamical system, whose stable equilibrium points are abstracted to a discrete set of "distinctive states." Trajectories linking these states can be abstracted to actions, giving a discrete causal graph level of representation for the state space. Depending on the properties of the actions, the causal graph can be deterministic or stochastic. The causal graph of states and actions can in turn be abstracted to a topological network of places and paths. Local metrical models, such as occupancy grids, of neighborhoods of places and paths can then be built on the framework of the topological network while avoiding their usual problems of global consistency. This paper gives an overview of the SSH, describes the kinds of guarantees that the representation can support, and gives examples from two different robot implementations. We conclude with a brief discussion of the relation between the concepts of "distinctive state" and "landmark value."