After a short introduction to some of the basic puzzles surrounding Quantum Mechanics I will study the effective quantum dynamics of systems interacting with a long chain of independent probes, which, afterwards, are subject to a projective measurement and are then lost. This leads us to develop a theory of indirect measurements of time-independent quantities (non-demolition measurements). Subsequently, the theory of indirect weak measurements of time-dependent quantities is outlined, and a new family of diffusion processes, quantum jump processes, is introduced. To conclude, some open problems are described.