  Many authors have studied sets, associated with the dynamics of a
transcendental entire function, which have the topological property of being a
spider's web. In this paper we adapt the definition of a spider's web to the
punctured plane. We give several characterisations of this topological
structure, and study the connection with the usual spider's web in
$\mathbb{C}$.
  We show that there are many transcendental self-maps of $\mathbb{C}^*$ for
which the Julia set is such a spider's web, and we construct a transcendental
self-map of $\mathbb{C}^*$ for which the escaping set $I(f)$ has this structure
and hence is connected. By way of contrast with transcendental entire
functions, we conjecture that there is no transcendental self-map of
$\mathbb{C}^*$ for which the fast escaping set $A(f)$ is such a spider's web.
