In this colloquium-style talk, we discuss how the Monge-Ampère equation arises in Kähler geometry, and motivate the search for sharp geometric estimates for it. A survey is then given of recent advances using auxiliary equations, including joint works of the speaker with B. Guo and F. Tong on C0 estimates, and with B. Guo, J. Song, and J. Sturm on the Green's function and applications to the Kähler-Ricci flow.