The majority of Earth's surface topography is controlled by variations in crustal and lithospheric thicknesses and densities that are principally generated at plate boundaries. Nevertheless, it is recognised that some fraction of topography is maintained by convective circulation of the Earth's interior. Geodynamic simulations typically predict that this dynamic component of surface topography varies on wavelengths of $\sim$ 10,000 km with amplitudes of up to $\sim$ $\pm$ 2 km. However, recent compilations of oceanic residual depth measurements reveal topographic anomalies on much shorter wavelengths of $\sim$ 1,000 km. Limited observational constraints exist regarding the spatial and temporal evolution of this dynamic topography on the continents. In this dissertation, I explore two ways in which such constraints might be obtained from fluvial landscapes and from sedimentary basins. I first investigate the growth of the Anatolian Plateaux. Elevated Miocene marine sedimentary deposits distributed across Anatolia indicate that significant uplift has occurred during Neogene and Quaternary times. However, the precise timing and spatial pattern of uplift are not well constrained. Fluvial drainage networks appear to contain quantitative information about the spatial and temporal evolution of continental topography. Such information is extracted from Anatolian drainage networks using an inverse modelling approach. Separately, estimates of asthenospheric temperature were obtained by modelling geochemical compositions of mafic volcanic rocks. Results suggest that elevated topography is supported by anomalously hot asthenosphere and by a thinned lithospheric plate. Regional uplift and magmatism appear to have propagated from east to west since $\sim$ 20 Ma. Consistent patterns linking uplift, magmatism and asthenospheric temperature imply that convective processes have had an important influence on the spatial and temporal evolution of the Anatolian Plateaux. Secondly, I focus on the Surprise Canyon Formation, a Carboniferous deposit that is exposed in the Grand Canyon of western North America. This formation, which contains terrestrial flora, consists of isolated channels incised into the underlying marine Redwall Limestone. These channels are overlain by the marine Supai Group. This stratigraphic relationship requires an episode of transient base-level fall. The chronology and evolution of the Surprise Canyon Formation is reviewed and revised, in order to compile a regional database of subsidence histories. Emplacement and subsequent cooling of warm asthenospheric material can explain significant stratigraphic relationships. Thirdly, I explore the structure and evolution of the intracratonic Congo sedimentary basin. This enigmatic basin's protracted stratigraphic record, which includes periods of rapid sedimentary acculation separated by long unconformities, remains poorly understood. Seismic reflection profiles acquired during the 1970s are re-analysed in order to improve the stratigraphic architecture. Subsidence histories for four wells are calculated and modelled. I show that these histories are consistent with basin initiation during a Neoproterozoic-Cambrian phase of rifiting followed by a long phase of thermal relaxation, a consequence of the basin's location on thick cratonic lithosphere. This post-rift phase is punctuated by periods of uplift and denudation that can be explained by emplacement and cooling of warm asthenospheric material at the base of the plate. I conclude that quantitative information about the dynamics of the Earth's mantle can be obtained by careful analysis of landscapes and the stratigraphic record.