  We study the dimension theory of limit sets of iterated function systems
consisting of a countably infinite number of contractions. Our primary focus is
on the intermediate dimensions: a family of dimensions depending on a parameter
$\theta \in [0,1]$ which interpolate between the Hausdorff and box dimensions.
Our main results are in the case when all the contractions are conformal. Under
a natural separation condition we prove that the intermediate dimensions of the
limit set are the maximum of the Hausdorff dimension of the limit set and the
intermediate dimensions of the set of fixed points of the contractions. This
builds on work of Mauldin and Urba\'nski concerning the Hausdorff and upper box
dimension. We give several (often counter-intuitive) applications of our work
to dimensions of projections, fractional Brownian images, and general H\"older
images. These applications apply to well-studied examples such as sets of
numbers which have real or complex continued fraction expansions with
restricted entries.
  We also obtain several results without assuming conformality or any
separation conditions. We prove general upper bounds for the Hausdorff, box and
intermediate dimensions of infinitely generated attractors in terms of a
topological pressure function. We also show that the limit set of a 'generic'
infinite iterated function system has box and intermediate dimensions equal to
the ambient spatial dimension, where 'generic' can mean either 'full measure'
or 'comeagre.'
