Hydrogen is receiving increasing attention as a versatile energy vector to help accelerate the transition to a decarbonised energy future. Gas turbines will continue to play a critical role in providing grid stability and resilience in future low-carbon power systems; however, it is recognised that this role is contingent upon achieving increased thermal efficiencies and the ability to operate on carbon-neutral fuels such as hydrogen. An important consideration in the development of gas turbine combustors capable of operating with pure hydrogen or hydrogen-enriched natural gas are the significant changes in thermoacoustic instability characteristics associated with burning these fuels. This article provides a review of the effects of burning hydrogen on combustion dynamics with focus on swirl-stabilised lean-premixed combustors. Experimental and numerical evidence suggests hydrogen can have either a stabilising or destabilising impact on the dynamic state of a combustor through its influence particularly on flame structure and flame position. Other operational considerations such as the effect of elevated pressure and piloting on combustion dynamics as well as recent developments in micromix burner technology for 100% hydrogen combustion have also been discussed. The insights provided in this review will aid the development of instability mitigation strategies for high hydrogen combustion.