  We review the motivation and fundamental properties of the Hausdorff
dimension of metric spaces and illustrate this with a number of examples, some
of which are expected and well-known. We also give examples where the Hausdorff
dimension has some surprising properties: we construct a set $E\subset\C$ of
positive planar measure and with dimension 2 such that each point in $E$ can be
joined to $\infty$ by one or several curves in $\C$ such that all curves are
disjoint from each other and from $E$, and so that their union has Hausdorff
dimension 1. We can even arrange things so that every point in $\C$ which is
not on one of these curves is in $E$. These examples have been discovered very
recently; they arise quite naturally in the context of complex dynamics, more
precisely in the iteration theory of simple maps such as $z\mapsto \pi\sin(z)$.
