For the first time, we present for the general case of fully nonlinear elliptic differential equations of second order on $C^2$ domains in $R^n$, a stability and convergence proof for a non standard non conforming $C^1$ finite element method and the variational crimes. Our proof is applicable to Davydov's $C^1$ finite elements on curved domains, available at the moment in $R^2$ and probably for $R^3$ soon. The general case of elliptic differential equations and systems of orders $2$ and $2m$ will not be discussed in the lecture. The method applies to non divergent quasilinear elliptic problems as well. Algorithms are formulated to calculate the nonlinear system and to solve it by a combination of continuation and discrete Newton methods. The latter converges locally quadratically, essentially independent of the actual grid size by the mesh independence principle. As usual for curved domains we have to consider the necessary quadrature and cubature approximations to avoid difficulties at the boundary. Essential tools are the interplay between the weak and strong form of the linearized operator and a new regularity result for solutions of finite element equations.