Modifying the photonic environment of a semiconductor quantum well by embedding it in a high-reflectivity microcavity gives rise to new fundamental optical excitations, half-quantum well excitons, half-photons. These particles, called polaritons, have a light mass, as cavity photons, meaning that they have a large De Broglie wavelength. On the other hand, polaritons, like excitons, are subject to Coulomb interaction, a feature generating strong optical nonlinearities. Such properties favour quantum degeneracy and collective phenomena related to the bosonic statistics of polaritons. We review experiments on stimulated scattering of polaritons. In particular we concentrate on the resonant excitation of polaritons somewhere on the dispersion curve and the stimulation of their scattering into the fundamental state by means of an optical probe beam. The process is called polariton parametric amplification and results in very large and ultrafast optical amplification of the probe beam. The model, based on a Hamiltonian of interacting bosons, suggests that the amplification is related to the coherence between polaritons. We demonstrate that in clearly designed samples, this coherence can be preserved almost up to room temperature, so that intersting applications of this phenomenon can be conceived. At the same time we have been able to improve dramatically the efficiency of the parametric process, making the microcavity an unprecedented optical amplifier.