To better understand the structure and function of complex systems, researchers often represent direct interactions between components in complex systems with networks, assuming that indirect influence between distant components can be modelled by paths. Such network models assume that actual paths are memoryless. That is, the way a path continues as it passes through a node does not depend on where it came from. Recent studies of data on actual paths in complex systems question this assumption and instead indicate that memory in paths does have considerable impact on central methods in network science. A growing research community working with so-called higher-order network models addresses this issue, seeking to take advantage of information that conventional network representations disregard. Here we summarise the progress in this area and outline remaining challenges calling for more research.