The dimensionality of the space has remarkable effects on the dynamics of turbulent flows. In three dimensions, the nonlinear interaction between different scales is described by the Kolmorogov-Richardson direct cascade: the kinetic energy injected at large scale by an external forcing is transferred to smaller and smaller eddies until it reaches the scales where it is dissipated by viscosity. By contrast, in two dimensions, the simultaneous conservation of kinetic energy and enstrophy results in an inverse energy cascade; i.e., the energy injected by the forcing is transferred to large-scale structure. In this lecture I will discuss the phenomenology of turbulent flows confined in thin fluid layers, which exhibit a transition from two-dimensional to three-dimensional dynamics as the thickness of the layer is increased. I will show that the transition is characterized by an intermediate regime in which both the direct and inverse cascades coexist. I will discuss the mechanisms which originate this phenomenon, focusing on the role played by inviscid invariants.