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We study germs of holomorphic mappings between general algebraic hypersurfaces. Our main result is the following. If $(M,p_0)$ and $(M^{\text{'}},p_0^{\text{'}})$ are two germs of real algebraic hypersurfaces in ${ℂ}^{N+1}$, $N\ge 1$, $M$ is not Levi-flat and $H$ is a germ at $p_0$ of a holomorphic mapping such that $H\left(M\right)\subseteq M^{\text{'}}$ and $\mathrm{Jac}\left(H\right)\not\equiv 0$ then the so-called reflection function associated to $H$ is always holomorphic algebraic. As a consequence, we obtain that if $M^{\text{'}}$ is given in the so-called normal form, the transversal component of $H$ is always algebraic. Another corollary of...

We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.