Following the general ideas  of  Fokas's Unified Transform Method, we are presenting an approach, based on the Riemann-Hilbert apparatus,  to the elastodynamic equations on the quarter plane.  The original boundary problem is reduced to amatrix Riemann-Hilbert problem with a shift posed on a tours. We shall also present the recent developments concerningthe exact calculation of the connection formulae between the Rayleigh waves propagatingalong vertical and horizontal boundaries. The talk is based on the earlier work of the speaker with A. Its and Yu.  Kaplunov and on the ongoing research  project with A. Its.