  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.
