Compact Kähler manifolds are natural generalizations of smooth projective varieties and particular examples of symplectic manifolds. In this talk, we will try to illustrate some pieces of frontiers between Kähler and algebraic geometry, as well as Kähler and symplectic geometry, by studying deformations of compact Kähler manifolds. For the former, we will focus on the Kodaira problem, asking whether every compact Kähler manifold admits arbitrarily small deformations to some projective manifolds. As for the latter, we will focus on the problem of finding properties coming from Kähler (or algebraic) geometry that are preserved under symplectic deformations.