Publication of the monograph : Scanning Brillouin microscopy: acoustic microscopy at gigahertz frequencies in the ‘Archives des Sciences naturelles, physiques et mathématiques’ édité par la Section des Sciences de l’Institut Grand-ducal. Abstract: Starting from acoustic microscopy, which bases on ultrasonic techniques, this document outlines the development of an alternative method, partially surpassing the classical acoustic microscopy: the scanning Brillouin microscopy. Since scanning Brillouin microscopy is an optical technique, it provides an entirely non-destructive access to the hypersonic velocity and hypersonic attenuation within transparent or translucent bulk samples under linear-response conditions. It is a striking feature of this technique that absolute values of hypersonic properties can be investigated. One-, two- or three-dimensional profiles of the acoustic or elastic inhomogeneities or heterogeneities can be generated for transparent materials. The properties of all acoustic modes can be recorded simultaneously, provided that the opto-acoustic coupling is sufficient. For non-transparent materials the acoustic properties of the surface, like surface acoustic waves, can be scanned. As Brillouin spectroscopy is an optical technique, the resolution of maps of the hypersonic properties is given by the optical diffraction limit, so that a lateral resolution of about 1 micron can be achieved. Hence, scanning Brillouin microscopy allows for combining the spatial, temporal and angular resolution of hypersonic properties. In other words, the complete information about the space- and time-dependence of the full elastic tensor can be resolved in transparent condensed matter. Several examples from the domains of soft and hard matter physics illustrate the worth of this experimental technique: the resolution of transcrystalline domains within a semi-crystalline polymer, the spatial analysis of mechanical interphases of epoxy-based adhesives deposited on different metal substrates, and the spatio-temporal characterisation of solvent uptake of glassy polymeric networks. The determination of the angle dependence of the three acoustic modes, in addition to their space dependence, delivers valuable insight into the spatial evolution of the elastic tensor of anisotropic inhomogeneous or heterogeneous matter. This will be demonstrated for uniaxially stretched polymer specimens and for concentration profiles of mixed crystals. Using polycrystalline diamond plates, it will be evidenced that scanning Brillouin microscopy provides a quantitative access to the spatial distribution of internal stress fields. It is the extremely low level of optical and acoustic perturbation imposed by the Brillouin technique which principally enables to resolve fragile structure formation processes, e.g. at unstable interfaces between liquids, in space and time.