In the present paper a generalized Kählerian space $\mathbb {G} {\underset 1 {\mathbb {K}}_N}$ of the first kind is considered as a generalized Riemannian space $\mathbb {GR}_N$ with almost complex structure $\smash {F^h_i}$ that is covariantly constant with respect to the first kind of covariant derivative. \endgraf Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings $f\colon \mathbb {GR}_N\to \mathbb {G}\underset 1{\mathbb {\overline {K}}}_N$ with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space $\mathbb {G}{\underset 1 {\mathbb {K}}}_N$.
In this paper we investigate holomorphically projective mappings of generalized Kählerian spaces. In the case of equitorsion holomorphically projective mappings of generalized Kählerian spaces we obtain five invariant geometric objects for these mappings.