Abstract: Quoting Kraichnan, the 2/3 and -5/3 power laws of Kolmogorov and Obukhov have « achieved an embarassment of success » as they have « been found not only where they reasonably could be expected » but also where the Kolmogorov theory is not designed for, e.g. in significantly non-homogeneous and near-field turbulence. The Kolmogorov theory of the turbulence cascade and scale-by-scale energy must therefore be a special case of a more general turbulence theory. To break beyond the Kolmogorov theory we need to break beyond its most constraining assumptions: equilibrium and homogeneity. Some progress has been made on non-equilibrium cascades and dissipation over the past ten years and so this talk’s focus is on non-homogeneity. A theory of non-homogeneous turbulence is developed and is applied to boundary-free shear flows. The theory introduces assumptions of inner and outer similarity for the non-homogeneity of two-point statistics and predicts power law scalings of second-order structure functions which have some similarities with but also some differences from Kolmogorov scalings. These scalings arise as a consequence of these assumptions, of the general inter-scale and inter-space energy balance and of an inner-outer equivalence hypothesis for turbulence dissipation. They reduce to usual Kolmogorov scalings in stationary homogeneous turbulence. Comparisons with structure function data from three qualitatively different turbulent wakes provide support for the theory’s predictions but also raise many new questions for future research.