The bending elasticity of a fluid membrane is characterized by its modulus and spontaneous curvature. We present a new method, advanced flicker spectroscopy of giant nonspherical vesicles, which makes it possible to simultaneously measure both parameters for the first time. Our analysis is based on the generation of a large set of reference data from Monte Carlo simulations of randomly triangulated surfaces. As an example of the potential of the procedure, we monitor thermal trajectories of vesicle shapes and discuss the elastic response of zwitterionic membranes to transmembrane pH gradients. Our technique makes it possible to easily characterize membrane curvature as a function of environmental conditions.