We provide an introduction to elasticity theory and related phenomena. The concept of strain and stress will be introduced, and the elastic constant matrix of the six crystal families and their stability will be discussed. We introduce concepts like Young’s modulus and Poisson ratio. As an examples of a static elasticity problem we will discuss the bending of a beam. 

We will review the dynamic equations of motion and their boundary condition; we evaluate the phonon dispersions and their dependence on crystallographic orientations. We discuss the method of ultrasound propagation and resonant ultrasound spectroscopy to determine the elastic constants. In addition, we analyze Rayleigh waves, i.e., surface acoustic waves (SAW). 

Moreover, we explore the coupling of quantum materials to elastic degrees of freedom. We introduce the observables expansivity, thermal expansion, magnetostriction, Grüneisen parameter and elastocaloric effect, and we discuss their typical temperature dependences. We review quantum phase transitions, i.e., instabilities of the ground state. Assuming a perturbative coupling of elastic and quantum critical degrees of freedom, we discuss the anomalous behaviour of observables close to quantum criticality. In addition, we analyze scenarios of a non-perturbative coupling. Finally, the concept of elastic quantum criticality, where the crystal lattice destabilizes at zero temperature, will be reviewed.