Other than being a natural first step after the study of canonical metrics on smooth manifolds, the study of canonical metrics on singular Kähler manifolds is motivated by the applications of such metrics in other branches of geometry, such as algebraic geometry and calibrated geometry. Every class of singularities poses certain technical difficulties. In this talk, I will concentrate on the case of the Calabi problem on Kähler manifold with isolated conical singularities, also known as conifolds. This covers, in particular, the case of Ricci-flat Kähler conifolds and allows generalizing the known results in the case of Calabi-Yau conifolds.