The report presents an exhaustive review of the recent attempt to overcomethe difficulties that standard quantum mechanics meets in accounting for themeasurement (or macro-objectification) problem, an attempt based on theconsideration of nonlinear and stochastic modifications of the Schroedingerequation. The proposed new dynamics is characterized by the feature of notcontradicting any known fact about microsystems and of accounting, on the basisof a unique, universal dynamical principle, for wavepacket reduction and forthe classical behavior of macroscopic systems. We recall the motivations forthe new approach and we briefly review the other proposals to circumvent theabove mentioned difficulties which appeared in the literature. In this way wemake clear the conceptual and historical context characterizing the newapproach. After having reviewed the mathematical techniques (stochasticdifferential calculus) which are essential for the rigorous and preciseformulation of the new dynamics, we discuss in great detail its implicationsand we stress its relevant conceptual achievements. The new proposal requiresalso to work out an appropriate interpretation; a procedure which leads us to areconsideration of many important issues about the conceptual status oftheories based on a genuinely Hilbert space description of natural processes.We also discuss the possibility and the problems one meets in trying to developan analogous formalism for the relativistic case. Finally we discuss theexperimental implications of the new dynamics for various physical processeswhich should allow, in principle, to test it against quantum mechanics.