Geostrophic turbulence is a key paradigm in the current understanding of the large-scale planetary circulations. It implies that a flow is turbulent, rotating, stably stratified, and in near-geostrophic balance. When a small-scale forcing is present, geostrophic turbulence features an inverse energy cascade. When the meridional variation of the Coriolis parameter (or a β-effect) is included, the horizontal flow symmetry breaks down giving rise to the emergence of jet flows. The presence of a large-scale drag ensures that the flow attains a steady state. Dependent on the governing parameters, four steady-state flow regimes are possible, two of which are considered in this study. In one of these regimes, a flow is dominated by the drag while in the other one, the recently discovered regime of zonostrophic turbulence, a flow becomes strongly anisotropic and features slowly evolving systems of alternating zonal jets. Zonostrophic turbulence is distinguished by anisotropic inverse energy cascade and emergence of a new class of nonlinear waves known as zonons. In addition, meridional scalar diffusion is strongly modified in this regime. This paper provides an overview of various regimes of turbulence with a β-effect, elaborates main characteristics of friction-dominated and zonostrophic turbulence, elucidates the physical nature of the zonons, discusses the meridional diffusion processes in different regimes, and relates these results to oceanic observations.