Spike train distances are rules for assigning a notion of distance, or dis-similarity, to pairs of event sequences. In contrast to most quantitative approaches to the analysis of scientific data, the framework of spike train distances does not make the implicit assumption that the objects of interest (i.e., the event sequences) can be thought of as vectors. In many cases, including the earliest examples of spike train distances (van Rossum 2001, Victor & Purpura 1996, Victor & Purpura 1997), these distances also satisfy the formal mathematical requirements to be a “metric” (see below). In this case, the event sequences may be considered elements in a “metric space.” A metric space is a topological space that is more general than a vector space. A metric space must have a notion of distance, but it need not have coordinates, nor allow for addition, scalar multiplication, or the measurement of angles. Two considerations give this general framework a special flavor when applied to neural data. The first consideration is mathematical: for event sequences, it is natural to think of the topology as combining a discrete component with a continuous component. The discrete component is that the number of events in a spike train must be an integer; the continuous component is that each of these events can occur across a continuum of times. The second consideration is biological: much is known about the physiology of neurons and neural circuits, and spike train distances are typically designed with the goal of capturing the biologically significant aspects of neuronal activity. As detailed below, two contrasting ideas concerning the biological meaning of a spike train serve as anchor points: (a) the firing events in a spike train might serve primarily as a means to represent an underlying firing rate vs. (b) the times of these firing events might have individual significance, enabling neural computations to be based on coincident firing events across neurons and other aspects of fine temporal structure.