D. Rhodri Davies, Siavash Ghelichkhan, Mark Hoggard, Andrew Valentine, Fred D. Richards

The slow creeping motion of Earth’s mantle drives transient changes in surface topography across a variety of spatial and temporal scales. Recent decades have seen substantial progress in understanding this so-called `dynamic topography’, with a growing number of studies highlighting its fundamental role in shaping the surface of our planet. In this review, we outline the current frontiers of geodynamical research into dynamic topography. It begins with a summary of ongoing observational, theoretical and computational efforts that aim to quantify the present-day expression of dynamic topography, including its geographical distribution and sensitivity to different components of the mantle's flow regime. Next, observational constraints that shed light on how dynamic topography has changed over time are summarised, and compared with predictions from a range of geodynamical modelling studies, to highlight our current understanding of its evolution through the geological past. Although many model predictions can be reconciled with the available observational constraints, these comparisons demonstrate that there remain inconsistencies, particularly at shorter spatial and temporal scales. These discrepancies allow us to isolate the shortcomings of existing modelling approaches and identify pathways towards improving future reconstructions of dynamic topography through space and time. Such reconstructions are vital if we are to robustly connect the evolution of Earth's surface environments to the processes that are occurring deep within its interior.