We give complete characterizations for the solvability of a class of  quasilinear and Hessian equations, which give a complete answer  to a 
problem posed by Bidaut-Veron.  As a result, we obtain a  characterization of removable singularities for the corresponding  homogeneous 
equations.  This is based on joint work with Igor E. Verbitsky  which will appear in the Annals of Mathematics.  The criteria of  solvability 
of analogous equations with general two weights are also  discussed.