Morelli and Wlodarczyk solved independently in 1996/97 the "weak Oda conjecture" for toric varieties. Stated geometrically, it says that any two triangulations of a convex polytope are related by geometric bistellar moves. In this talk I will give a version of a proof of the Morelli-Wlodarczyk theorem and discuss two applications. The first application is the rigidity of a certain class of non-convex polyhedra (joint work with Jean-Marc Schlenker). The second is is a proof of Billera's conjecture: any simplicial PL sphere becomes polytopal after several derived subdivisions (joint work with Karim Adiprasito).