  In its most restrictive definition, an octupolar tensor is a fully symmetric
traceless third-rank tensor in three space dimensions. So great a body of works
have been devoted to this specific class of tensors and their physical
applications that a review would perhaps be welcome by a number of students.
Here, we endeavour to place octupolar tensors into a broader perspective,
considering non-vanishing traces and non-fully symmetric tensors as well. A
number of general concepts are recalled and applied to either octupolar and
higher-rank tensors. As a tool to navigate the diversity of scenarios we
envision, we introduce the octupolar potential, a scalar-valued function which
can easily be given an instructive geometrical representation. Physical
applications are plenty; those to liquid crystal science play a major role
here, as they were the original motivation for our interest in the topic of
this review.
