Models of spontaneous wave function collapse make predictions, which are different from those of standard quantum mechanics. Indeed, these models can be considered as a rival theory, against which the standard theory can be tested, in pretty much the same way in which parametrized post-Newtonian gravitational theories are rival theories of general relativity. The predictions of collapse models almost coincide with those of standard quantum mechanics at the microscopic level, as these models have to account for the microscopic world, as we know it. Departures become significant when the size of the system increases. However, for larger systems environmental influences become more and more difficult to eliminate. This is the reason why it is tricky to test collapse models experimentally, and so far no decisive test has been performed. We will review the main phenomenological properties of collapse models, in particular the so-called amplification mechanics, as well as the main models, which are debated in the literature (GRW, CSL, QMUPL, DP). We will review the lower bounds on the collapse parameter, and more importantly the upper bounds set by available experimental data. This data come both from experimental tests on earth, and from cosmological observations.