A laser-based technique for the contact-free generation and detection of strongly nonlinear surface acoustic wave (SAW) pulses with amplitudes limited by the materials strength has been developed. The effects of nonlinear propagation of short elastic surface pulses with finite strength in isotropic solids, such as fused quartz, anisotropic solids, such as silicon, and dispersive media were investigated. Solitary surface wave propagation was observed in layered structures for normal and anomalous dispersion. In addition, a SAW-based method for evaluating the critical fracture stress of anisotropic brittle solids, such as single crystal silicon, is introduced.