In this review paper, we illustrate a possible route to obtain a reliable solution of the 2D Hubbard model and an explanation for some of the unconventional behaviours of underdoped high-T_c cuprate superconductors within the framework of the composite operator method. The latter is described exhaustively in its fundamental philosophy, various ingredients and robust machinery to clarify the reasons behind its successful applications to many diverse strongly correlated systems, controversial phenomenologies and puzzling materials.