Convexity is a crucial topic in the analysis of partial differential equations. In this talk, we present our work on the convexity of solutions to complex Monge-Ampère equations. We begin by discussing geodesics in the space of Kähler potentials, which motivates our study of the homogeneous complex Monge-Ampère equation. Next, we present results on the preservation of convexity properties for the homogenous complex Monge-Ampère equation, followed by new findings for the non-homogeneous equation.