  We establish $C^2$ a priori estimate for convex hypersurfaces whose principal
curvatures $\kappa=(\kappa_1,..., \kappa_n)$ satisfying Weingarten curvature
equation $\sigma_k(\kappa(X))=f(X,\nu(X))$. We also obtain such estimate for
admissible 2-convex hypersurfaces in the case $k=2$. Our estimates resolve a
longstanding problem in geometric fully nonlinear elliptic equations.
