Elastic properties of nanoscopic materials, structures and thin films are important parameters controlling their growth, as well as their optical and electronic properties. Acoustic microscopy is a well-established method for elastic imaging. In order to overcome its micrometer-scale diffraction-limited lateral resolution, scanning probe microscopy-based acoustic near-field techniques have been developed. Among the acoustic modes used for microscopy, surface acoustic waves (SAWs) are especially suited for probing very small and thin objects due to their localization in the vicinity of the surface. Moreover, the study of SAWs is crucial for the design of frequency filter devices as well as for fundamental physical studies, for instance, the probing of composite fermions in two-dimensional electron systems. This review discusses the capabilities and limitations of SAW-based scanning probe microscopy techniques. Particular emphasis is laid on the review of surface acoustic waves and their interaction with elastic inhomogeneities. Scattering, diffraction and wave localization phenomena will be discussed in detail. Finally, the possibilities for quantitative acoustic microscopy of objects on the nanoscale, as well as practical applications, are presented.