We give an introduction into the theory of neutron scattering in condensed matter with a strong focus on inelastic scattering. Magnetic scattering and polarisation will be excluded from the discussion. The scattering of non-relativistic particles in a potential leads to the concepts of scattering amplitude, partial waves, scattering length, and the Born series. In order to illustrate the formalism with simple examples we allow ourselves a short detour to diffraction. The Born approximation is then augmented to include inelastic scattering from compounds with internal degrees of freedom. This leads to the master equation of neutron scattering that establishes a link between the experimental cross sections and quantum mechanical transition amplitudes. The master equation is reformulated in terms of coherent and incoherent scattering functions, which in turn are expressed in terms of density correlation functions. The abstract formalism is illustrated by explicitly calculating the cross sections for some simple model systems. Considerable attention is given to the scattering involving vibrations in harmonic systems. Explicit expressions are derived that couple inelastic scattering cross sections to the phonon density of states and dispersion relations. We will finish the inelastic part by discussing the effect of multi-phonons and conclude with some remarks concerning the identification of anharmonic effects. The last section is devoted to putting the formalism of scattering theory into the context of a statistical description of the particle beam.