We consider the inhomgeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity. Initially the fluids are supposed to be at rest and separated by a flat horizontal interface with the heavier fluid being on top of the lighter one. Due to gravity this configuration is unstable, the two fluids begin to mix in a more and more turbulent way. This is called the Rayleigh-Taylor instability. In the talk we will see how admissible solutions to the Euler equations reflecting a turbulent mixing of the two fluids in a quadratically growing zone can be constructed. Furthermore, if time allows, we will discuss an arising selection problem for the averaged motion of solutions. This is based on joint works with József Kolumbán, László Székelyhidi, Jonas Hirsch.