A disturbance in the local magnetic order of a solid body can propagate across a material just like a wave. This wave is named spin wave, and its quanta are known as magnons. Recently, physicists proposed the usage of magnons to carry and process information instead of electrons as it is the case of electronics. This technology opens access to a new generation of computers in which data are processed without motion of any real particles like electrons. This leads to a sizable decrease in the accompanying heating losses and, consequently, to lower energy consumption, which is crucial due to the ever increasing demand for computing devices. Moreover, unique properties of spin waves allow for the utilisation of unconventional computing concepts, giving the vision of a significantly faster and more powerful next-generation of information processing systems. The current review addresses a selection of fundamental topics that form the basis of the magnon-based computing and are of primary importance for the further development of this concept. First, the transport of spin-wave-carried information in one and two dimensions that is required for the realization of logic elements and integrated magnon circuits is covered. Second, the convertors between spin waves and electron (charge and spin) currents are discussed. These convertors are necessary for the compatibility of magnonic devices with modern CMOS technology. The paper starts with basics on spin waves and the related methodology. In addition, the general ideas behind magnon-based computing are presented. The review finishes with conclusions and an outlook on the perspective use of spin waves.