  We derive a general upper bound for the number of incidences with
$k$-dimensional varieties in ${\mathbb R}^d$. The leading term of this new
bound generalizes previous bounds for the special cases of $k=1, k=d-1,$ and
$k= d/2$, to every $1\le k <d$. We derive lower bounds showing that this
leading term is tight in various cases. We derive a bound for incidences with
transverse varieties, generalizing a result of Solymosi and Tao. Finally, we
derive a bound for incidences with hyperplanes in ${\mathbb C}^d$, which is
also tight in some cases. (In both ${\mathbb R}^d$ and ${\mathbb C}^d$, the
bounds are tight up to sub-polynomial factors.)
  To prove our incidence bounds, we define the \emph{dimension ratio} of an
incidence problem. This ratio provides an intuitive approach for deriving
incidence bounds and isolating the main difficulties in each proof. We rely on
the dimension ratio both in ${\mathbb R}^d$ and in ${\mathbb C}^d$, and also in
some of our lower bounds.
