In joint work with K. Zhang we construct some explicit canonical geometries on various classes of complex manifolds, following a general symmetry principle pioneered by Calabi in the 70's. Our focus is to allow edge type singularities (that are the natural higher-dimensional analogues of conical Riemann surfaces studied by Picard and others since the 19th century) and study Gromov-Hausdorff limits as the angle in the cone tends to zero. In the process we also recover many classical smooth (i.e., without singularities) constructions by Calabi himself, as well as Cao, Futaki, Hamilton, Ni, and others. Some of these constructions, while very elementary, are new already in the Riemann surface case. We also confirm a conjecture of Cheltsov and myself on Calabi-Yau fibrations in some very particular cases.