I will review key computational aspects of navigation and describe the brain’s circuits that underlie it. These circuits contain neurons with remarkable responses to spatial variables, including head direction cells, grid cells, and place cells. I’ll describe the likely unifying mechanism for memory and (vector) integration of velocity signals in some of these circuits, across species, including support for the models from quantitative analysis of neural data. I’ll describe how it can be possible to discover, simply from the time-series of spikes in a neural population, the unknown variables a circuit encodes. Finally, given that spatial computations unfold over time, I will review theoretical and experimental efforts to understand the temporal dynamics of navigational computations. I will mention open questions throughout.