  We study incidence problems involving points and curves in $R^3$. The current
(and in fact only viable) approach to such problems, pioneered by Guth and
Katz, requires a variety of tools from algebraic geometry, most notably (i) the
polynomial partitioning technique, and (ii) the study of algebraic surfaces
that are ruled by lines or, in more recent studies, by algebraic curves of some
constant degree. By exploiting and refining these tools, we obtain new and
improved bounds for point-curve incidence problems in $R^3$.
  Incidences of this kind have been considered in several previous studies,
starting with Guth and Katz's work on points and lines. Our results, which are
based on the work of Guth and Zahl concerning surfaces that are doubly ruled by
curves, provide a grand generalization of most of the previous results. We
reconstruct the bound for points and lines, and improve, in certain significant
ways, recent bounds involving points and circles (in Sharir, Sheffer and Zahl),
and points and arbitrary constant-degree algebraic curves (in Sharir, Sheffer
and Solomon). While in these latter instances the bounds are not known (and are
strongly suspected not) to be tight, our bounds are, in a certain sense, the
best that can be obtained with this approach, given the current state of
knowledge.
  As an application of our point-curve incidence bound, we show that the number
of triangles spanned by a set of $n$ points in $R^3$ and similar to a given
triangle is $O(n^{15/7})$, which improves the bound of Agarwal et al. Our
results are also related to a study by Guth et al.~(work in progress), and have
been recently applied in Sharir, Solomon and Zlydenko to related incidence
problems in three dimensions.
