Integrodifference equations provide a very natural general framework to model the spread of invasive species if the species in question has a clearly distinct growth and dispersal phase during its life cycle. Many insect species satisfy this description, in particular where climate imposes strong seasonality. Early applications of integrodifference equations considered homogeneous landscapes and a number of relatively simple dispersal mechanisms. I will report on some recent developments concerning heterogeneous landscapes and density-dependent dispersal. I will explain several approximations that can be used to identify important spatial scales and simplify model parametrization. The main focus of the presentation will be on spreading speeds, and, in the case of landscape heterogeneity, also on persistence conditions. Several insights about landscape alterations for spread control will be discussed. I will end with some open and challenging questions for integrodifference equations as applied to invasive species.