The classical Kodaira embedding theorem asserts that a compact Kähler manifold with positive Ricci curvature is necessarily projective. Various generalizations of this theorem have been proposed for Kähler manifolds satisfying various positive or quasi-positive curvature conditions. In this lecture, I will first introduce some new results related to the projectivity of Kähler manifolds under certain quasi-positive curvature conditions, including quasi-positive $S_2^\perp,  S_2^+,\mbox{Ric}_3^\perp, \mbox{Ric}_3^+$ or $2$-quasi-positive $\mbox{Ric}_k$. Subsequently, I will demonstate that a compact K\"ahler manifold with a special holonomy group is also projective if it satisfies certain non-negative curvature condition, including non-negative  $S_2^\perp, S_2^+,\mbox{Ric}_3^\perp, \mbox{Ric}_3^+$ or $2$-non-negative $\mbox{Ric}_k$. This is a joint work with Yiyang Du.