The elastoresistivity tensor m_{ij,kl} characterizes changes in a material's resistivity due to strain. As a fourth-rank tensor, elastoresistivity can be a uniquely useful probe of the symmetries and character of the electronic state of a solid. We present a symmetry analysis of m_{ij,kl} (both in the presence and absence of a magnetic field) based on the crystalline point group, focusing for pedagogic purposes on the D_{4h} point group (of relevance to several materials of current interest). We also discuss the relation between m_{ij,kl} and various thermodynamic susceptibilities, particularly where they are sensitive to critical fluctuations proximate to a critical point at which a point group symmetry is spontaneously broken. Show Partial Abstract