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We study a wide spectrum of incidence problems involving points and curves orpoints and surfaces in $\mathbb R^3$. The current (and in fact the only viable)approach to such problems, pioneered by Guth and Katz [2010,2015], requires avariety of tools from algebraic geometry, most notably (i) the polynomialpartitioning technique, and (ii) the study of algebraic surfaces that are ruledby lines or, in more recent studies [Guth-Zahl 2016], by algebraic curves ofsome constant degree. By exploiting and refining these tools, we obtain new andimproved bounds for numerous incidence problems in $\mathbb R^3$. In broadterms, we consider two kinds of problems, those involving points andconstant-degree algebraic \emph{curves}, and those involving points andconstant-degree algebraic \emph{surfaces}. In some variants we assume that thepoints lie on some fixed constant-degree algebraic variety, and in others weconsider arbitrary sets of points in 3-space. our results provide a "grandgeneralization" of most of the previous studies of (special instances of)previous works for both curves and surfaces.  As an application of our point-curve incidence bound, we consider the problemof bounding the number of similar triangles spanned by a set of $n$ points in$\mathbb R^3$, and obtain the bound $O(n^{15/7}),$ thus improving the bound ofAgarwal et al [2007].  As applications of our point-surface incidence bounds, we consider theproblems of distinct and repeated distances determined by a set of $n$ pointsin $\mathbb R^3$, two of the most celebrated open problems in combinatorialgeometry. We obtain new and improved bounds for two special cases, one in whichthe points lie on some algebraic variety of constant degree, and one involvingincidences between pairs in $P_1\times P_2$, where $P_1$ is contained in avariety and $P_2$ is arbitrary.

Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances