Perhaps the simplest and most complete description of the cerebral cortex is that it is a sensorimotor controller whose primary purpose is to represent stimuli and movements, and adaptively control the mapping between them. However, in order to think, the cerebral cortex has to generate patterns of neuronal activity that encode abstract, generalized information independently of ongoing sensorimotor events. A critical question confronting cognitive systems neuroscience at present therefore is how neural signals encoding abstract information emerge within the sensorimotor control networks of the brain. In this review, we approach that question in the context of the neural representation of space in posterior parietal cortex of non-human primates. We describe evidence indicating that parietal cortex generates a hierarchy of spatial representations with three basic levels: including (1) sensorimotor signals that are tightly coupled to stimuli or movements, (2) sensorimotor signals modified in strength or timing to mediate cognition (examples include attention, working memory, and decision-processing), as well as (3) signals that encode frankly abstract spatial information (such as spatial relationships or categories) generalizing across a wide diversity of specific stimulus conditions. Here we summarize the evidence for this hierarchy, and consider data showing that signals at higher levels derive from signals at lower levels. That in turn could help characterize neural mechanisms that derive a capacity for abstraction from sensorimotor experience.