In the present Thesis, we deal mainly with the subclass of Hamiltonian evolutionary PDEs and their Poisson brackets (PBs). We address the problem of classifying discrete differential-geometric PBs (dDGPBs) of any \ufb01xed order on target space of dimension 1 and describing their compatible pencils. Furthermore, we explain a new criterium about the existence of tri-Hamiltonian structures for nonlinear wave systems
