Abstract
Magnetic waves are a relevant component in the dynamics of the solar atmosphere. Their significance has increased because of their potential as a remote diagnostic tool and their presumed contribution to plasma heating processes. We discuss our current understanding of coronal heating by magnetic waves, based on recent observational evidence and theoretical advances. The discussion starts with a selection of observational discoveries that have brought magnetic waves to the forefront of the coronal heating discussion. Then, our theoretical understanding of the nature and properties of the observed waves and the physical processes that have been proposed to explain observations are described. Particular attention is given to the sequence of processes that link observed wave characteristics with concealed energy transport, dissipation and heat conversion. We conclude with a commentary on how the combination of theory and observations should help us to understand and quantify magnetic wave heating of the solar atmosphere.