Evolutionary adaptation is a key driver of species' range dynamics. Understanding the factors that affect rates of adaptation at range margins is thus crucial for interpreting and predicting changes in species' ranges. The spatial structure of environmental conditions is one of the determinants of whether and how quickly adaptations occur. However, while landscape structures at range edges are typically complex, most theoretical work has so far focused on relatively simple environmental geometries. Using an individual-based allelic model, we explore the effects of different landscape structures on the rate of adaptation to novel environments and investigate how these structures interact with the genetic architecture of the trait governing adaptation and the dispersal capacity of the considered species. Generally, we find that rapid adaptation is favored by a good match between the coarseness of the trait's genetic architecture (many loci of small effects versus few loci of large effects) and the coarseness of the landscape (abruptness of transitions in environmental conditions). For example, in rugged landscapes, adaptation is quicker for genetic architectures with few loci of large effects, while for shallow gradients the opposite is true. Moreover, dispersal capacities affect the rate of adaptation by modulating the 'apparent coarseness' of the landscape: a gradient perceived as smooth by species with limited dispersal capacities appears rather steep for highly dispersive ones. We also find that the distribution of evolving phenotypes strongly depends on the interplay of landscape structure and dispersal capacities, ranging from two distinct phenotypes for most rugged landscapes, over the co-occurrence of an additional third phenotype for highly dispersive species, to the whole range of phenotypes on smooth gradients. By identifying basic factors that drive the fixation probability of newly arising beneficial mutations, we hope to further broaden the understanding of evolutionary adaptation at range margins and, hence, species' range dynamics.