The problem of prescribing the k-th elementary symmetric functions of the Schouten tensor in a conformal class is a natural fully nonlinear analogue of the Yamabe Problem.  In this talk, I describe a family of conformally invariant boundary operators which are naturally adapted studying these problems on manifolds with boundary; these operators prescribe S. Chen's H_k-curvatures of the boundary and enable one to generalize work of Escobar for the scalar curvature.  One application is a Dirichlet principle for extending a metric on the boundary to one in the interior for which sigma_k vanishes.  Another application is the construction of conformal classes constaining multiple metrics with vanishing sigma_k and H_k-curvature constant equal to one.  This is joint work with Yi Wang and Ana Claudia Moreira.