  The heating of the solar chromosphere and corona to the observed high
temperatures, imply the presence of ongoing heating that balances the strong
radiative and thermal conduction losses expected in the solar atmosphere. It
has been theorized for decades that the required heating mechanisms of the
chromospheric and coronal parts of the active regions, quiet-Sun, and coronal
holes are associated with the solar magnetic fields. However, the exact
physical process that transport and dissipate the magnetic energy which
ultimately leads to the solar plasma heating are not yet fully understood. The
current understanding of coronal heating relies on two main mechanism:
reconnection and MHD waves that may have various degrees of importance in
different coronal regions. In this review we focus on recent advances in our
understanding of MHD wave heating mechanisms. First, we focus on giving an
overview of observational results, where we show that different wave modes have
been discovered in the corona in the last decade, many of which are associated
with a significant energy flux, either generated in situ or pumped from the
lower solar atmosphere. Afterwards, we summarise the recent findings of
numerical modelling of waves, motivated by the observational results. Despite
the advances, only 3D MHD models with Alfv\'en wave heating in an unstructured
corona can explain the observed coronal temperatures compatible with the quiet
Sun, while 3D MHD wave heating models including cross-field density structuring
are not yet able to account for the heating of coronal loops in active regions
to their observed temperature.
