  We establish an improved upper bound for the number of incidences between m
points and n circles in three dimensions. The previous best known bound,
originally established for the planar case and later extended to any dimension
$\ge 2$, is $O*(m^{2/3}n^{2/3} + m^{6/11}n^{9/11}+m+n)$, where the $O*(\cdot)$
notation hides sub-polynomial factors. Since all the points and circles may lie
on a common plane (or sphere), it is impossible to improve the bound in R^3
without first improving it in the plane.
  Nevertheless, we show that if the set of circles is required to be "truly
three-dimensional" in the sense that no sphere or plane contains more than $q$
of the circles, for some $q << n$, then the bound can be improved to
\[O*(m^{3/7}n^{6/7} + m^{2/3}n^{1/2}q^{1/6} + m^{6/11}n^{15/22}q^{3/22} + m +
n). \]
  For various ranges of parameters (e.g., when $m=\Theta(n)$ and $q =
o(n^{7/9})$), this bound is smaller than the lower bound
$\Omega*(m^{2/3}n^{2/3}+m+n)$, which holds in two dimensions.
  We present several extensions and applications of the new bound: (i) For the
special case where all the circles have the same radius, we obtain the improved
bound $O*(m^{5/11}n^{9/11} + m^{2/3}n^{1/2}q^{1/6} + m + n$. (ii) We present an
improved analysis that removes the subpolynomial factors from the bound when
$m=O(n^{3/2-\eps})$ for any fixed $\varepsilon >0$. (iii) We use our results to
obtain the improved bound $O(m^{15/7})$ for the number of mutually similar
triangles determined by any set of $m$ points in R^3.
  Our result is obtained by applying the polynomial partitioning technique of
Guth and Katz using a constant-degree partitioning polynomial (as was also
recently used by Solymosi and Tao). We also rely on various additional tools
from analytic, algebraic, and combinatorial geometry.
