Given a super-Riemann surface defined by a Fuchsian group Gamma , the global space of super-Beltrami differential, parametrising deformations of the given superconformal structure, is defined and studied. The method (using vector fields satisfying an integrability condition) simplifies the definition considerably, and leads to natural analogues of the basic results in Beltrami theory. In particular the author is able to state and solve the super-Beltrami equations, and to prove the global existence of the Bers embedding for super-Teichmuller space. These results clarify previous work of Crane, Rabin and others (1986) and provide a foundation for the study of the supermoduli space in genus >1.