Inelastic scattering of X-rays with very high energyresolution has finally become possible thanks to a newgeneration of high-intensity X-ray sources.This development marks the end to the traditional beliefthat low energy excitations like lattice vibrations cannotbe resolved directly with X-rays: Inelastic scatteringexperiments allow to observe directly the small energyshifts of the photons. Studies of lattice vibrations, ofexcitations in molecular crystals, of collective excitationsin liquids and electronic excitations in crystalsdemonstrating the broad applicability and power of this newtechnology are discussed in this book.The progress in this field opens up fantastic new researchareas not only in physics but also in other disciplines suchas materials science,biology and chemistry.