We study the most general supersymmetric warped M-theory backgrounds with a non-trivial G flux of the type R 1, 2× M 8 and AdS 3× M 8. We give a set of necessary and sufficient conditions for preservation of supersymmetry which are phrased in terms of G structures and their intrinsic torsion. These equations may be interpreted as calibration conditions for a static “dyonic” M-brane, that is, an M5-brane with a self-dual three-form turned on. When the electric flux is turned off we obtain the supersymmetry conditions and non-linear PDEs describing M5-branes wrapped on associative and special Lagrangian three-cycles in manifolds with G 2 and SU (3) structures, respectively. As an illustration of our formalism, we recover the 1/2 BPS dyonic M-brane, and also construct some new examples.