  We consider the Banach space consisting of continuous functions from an
arbitrary uncountable compact metric space, $X$, into $\mathbb{R}^n$. The key
question is `what is the generic dimension of $f(X)$?' and we consider two
different approaches to answering it: Baire category and prevalence. In the
Baire category setting we prove that typically the packing and upper box
dimensions are as large as possible, $n$, but find that the behaviour of the
Hausdorff, lower box and topological dimensions is considerably more subtle. In
fact, they are typically equal to the minimum of $n$ and the topological
dimension of $X$. We also study the typical Hausdorff and packing measures of
$f(X)$ and, in particular, give necessary and sufficient conditions for them to
be zero, positive and finite, or infinite.
  It is interesting to compare the Baire category results with results in the
prevalence setting. As such we also discuss a result of Dougherty on the
prevalent topological dimension of $f(X)$ and give some simple applications
concerning the prevalent dimensions of graphs of real-valued continuous
functions on compact metric spaces, allowing us to extend a recent result of
Bayart and Heurteaux.
