Superfluid turbulence, also known as quantum turbulence, is characterised by the existence of quantum vortices that are topological defects of the order parameter describing the system. Typical examples are superfluid Helium 4 and Bose-Einstein condensates. At large scales, superfluids exhibit a classical turbulent Kolmogorov energy cascade, whereas at small scales, the energy is transferred by non-linear waves propagating along vortex filaments. These vortical excitations are known as Kelvin waves and the physics is described by the theory of weak-wave turbulence. Different models are used to describe superfluids. In this talk, I will focus on the 3D Gross-Pitaevskii equation (also known as the de-focusing NLS equation) that naturally contains turbulent quantum vortices, sound and Kelvin waves. After an introduction and explaining the different waves coexisting with these vortices, I will review the Kelvin wave cascade predictions and present numerical results confirming the existence of this cascade in a turbulent tangle.