Monin-Obukhov similarity theory (MOST) forms the basis for parametrizations of turbulent exchange in virtually all numerical models of atmospheric flows. Yet, its limitations to flat and horizontally homogeneous terrain have plagued the theory since its inception. Here we present a first generalized extension of MOST based on the inclusion of turbulence anisotropy as an additional nondimensional term. This novel theory developed based on an unprecedented ensemble of complex atmospheric turbulence datasets covering flat to mountainous terrain, is shown to be valid in conditions in which MOST fails and thus paves the way to a better understanding of complex turbulence.