  Exact budget equations for the second-order structure function tensor
$\langle \delta u_i \delta u_j \rangle$ are used to study the two-point
statistics of velocity fluctuations in inhomogeneous turbulence. The
Anisotropic Generalized Kolmogorov Equations (AGKE) describe the production,
transport, redistribution and dissipation of every Reynolds stress component
occurring simultaneously among different scales and in space, i.e. along
directions of statistical inhomogeneity. The AGKE are effective to study the
inter-component and multi-scale processes of turbulence. In contrast to more
classic approaches, such as those based on the spectral decomposition of the
velocity field, the AGKE provide a natural definition of scales in the
inhomogeneous directions, and describe fluxes across such scales too. Compared
to the Generalized Kolmogorov Equation, which is recovered as their half trace,
the AGKE can describe inter-component energy transfers occurring via the
pressure-strain term and contain also budget equations for the off-diagonal
components of $\langle \delta u_i \delta u_j \rangle$.
  The non-trivial physical interpretation of the AGKE terms is demonstrated
with three examples. First, the near-wall cycle of a turbulent channel flow at
$Re_\tau=200$ is considered. The off-diagonal component $\langle -\delta u
\delta v \rangle$, which can not be interpreted in terms of scale energy, is
discussed in detail. Wall-normal scales in the outer turbulence cycle are then
discussed by applying the AGKE to channel flows at $Re_\tau=500$ and $1000$. In
a third example, the AGKE are computed for a separating and reattaching flow.
The process of spanwise-vortex formation in the reverse boundary layer within
the separation bubble is discussed for the first time.
