1. Introduction -- 2. Surface Acoustic Waves -- 2.1 Surface Acoustic Waves on Various Media -- 2.2 Generation and Detection of Surface Acoustic Waves -- 2.3 Some Applications of Surface Acoustic Waves -- References -- 3. The Green's Function Method in the Surface Lattice Dynamics of Ionic Crystals -- 3.1 Outline of the Time-Independent Green's Function Method -- 3.2 The Green's Function Method in Surface Dynamics -- 3.3 The Intrinsic Perturbation for a Semi-Infinite Lattice -- 3.4 The Electronic Contribution to Surface Dynamics in the Framework of Shell Models -- 3.5 Surface Phonon Polaritons -- 3.6 Surface Vibrations in Alkali Halides -- 3.7 Further Developments: The Study of Surface Phonon Anomalies -- References -- 4. Study of Surface Phonons by the Slab Method -- 4.1 Formalism -- 4.2 Computational Considerations -- 4.3 Interaction Models -- 4.4 Results -- 4.5 Derived Physical Quantities -- 4.6 Concluding Remarks -- References -- 5. Experimental Determination of Surface Phonons by Helium Atom and Electron Energy Loss Spectroscopy -- 5.1 Theoretical Background -- 5.2 Kinematics -- 5.3 Helium Scattering -- 5.4 Electron Scattering -- 5.5 Intensities -- 5.6 Discussion of Experimental Results and Summary -- References -- 6. Theory of Helium Scattering from Surface Phonons -- 6.1 Kinematics -- 6.2 Dynamical Theory: General Considerations -- 6.3 One-Phonon Exchange Processes -- 6.4 The Inelastic Atom-Surface Interaction -- References -- 7. The Study of Surface Phonons by Electron Energy Loss Spectroscopy: Theoretical and Experimental Considerations -- 7.1 A Brief Review -- 7.2 The Surface Phonon Excitation Mechanism in the Impact Regime -- 7.3 The Green's Function Approach to Spectral Density Calculations -- 7.4 Calculations of the Cross Section for Surface Phonon Excitation -- 7.5 Concluding Remarks -- References -- 8. Vibrational Properties of Clean Surfaces: Survey of Recent Theoretical and Experimental Results -- 8.1 Ionic Crystals -- 8.2 Metals -- 8.3 Miscellaneous -- References.