The most urgent problems in relation to surface wave analysis and applications, which are comprehensively discussed here, are: Development of an adequate theory to analyze the "forbidden" direction problem for genuine surface waves; analyzing nonclassical surface waves propagating in forbidden directions; development of efficient numerical methods and algorithms to analyze surface waves (including Love and Lamb waves) propagating in homogeneous and layered media with both arbitrary elastic anisotropy and plasticity and having a complex internal structure; development of experimental and theoretical procedures to identify material properties, and solitary and dispersed defects by non-destructive testing; development of efficient analytical and numerical methods to analyze surface waves in porous, water-saturated media and ice fields; development of analytical and numerical methods to analyze interactions of cracks, faults, step discontinuities and edges with surface waves; improving the theory of crack propagation in relation to the analysis of surface wave velocities; and developing a theory to predict the behaviour of nonlinear surface waves.