CR Extension from Manifolds of Higher Type This paper deals with the extension of CR functions from a manifold $M\subset \mathbb C^n$ into directions produced by higher order commutators of holomorphic and antiholomorphic vector fields. It uses the theory of complex sectors'' attached to real submanifolds introduced in recent joint work of the authors with D. Zaitsev. In addition, it develops a new technique of approximation of sectors by smooth discs. Categories:32V25, 32V35, 32C16, 32F18