  We introduce a continuum of dimensions which are `intermediate' between the
familiar Hausdorff and box dimensions. This is done by restricting the families
of allowable covers in the definition of Hausdorff dimension by insisting that
$|U| \leq |V|^\theta$ for all sets $U, V$ used in a particular cover, where
$\theta \in [0,1]$ is a parameter. Thus, when $\theta=1$ only covers using sets
of the same size are allowable, and we recover the box dimensions, and when
$\theta=0$ there are no restrictions, and we recover Hausdorff dimension.
  We investigate many properties of the intermediate dimension (as a function
of $\theta$), including proving that it is continuous on $(0,1]$ but not
necessarily continuous at $0$, as well as establishing appropriate analogues of
the mass distribution principle, Frostman's lemma, and the dimension formulae
for products. We also compute, or estimate, the intermediate dimensions of some
familiar sets, including sequences formed by negative powers of integers, and
Bedford-McMullen carpets.
