The heating of the solar chromosphere and corona to the observed high temperatures, imply the presence of continuous heating that balances the strong radiative and thermal conduction losses expected in the solar atmosphere. It was theorized for decades that the required heating mechanisms of the chromosphere, active regions and quiet Sun 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 have not yet been fully understood. The current understanding of coronal heating relies on two main mechanism: recon- nection 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. Afterwards, we summarise the recent findings of numerical modelling, motivated by the observational results. Despite the advances, only 3D MHD models with Alfvén 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.