Insect navigation is strikingly geometric. Many species use path integration to maintain an accurate estimate of their distance and direction (a vector) to their nest and can store the vector information for multiple salient locations in the world, such as food sources, in a common coordinate system. Insects can also use remembered views of the terrain around salient locations or along travelled routes to guide return, which is a fundamentally geometric process. Recent modelling of these abilities shows convergence on a small set of algorithms and assumptions that appear sufficient to account for a wide range of behavioural data. Notably, this 'base model' does not include any significant topological knowledge: the insect does not need to recover the information (implicit in their vector memory) about the relationships between salient places; nor to maintain any connectedness or ordering information between view memories; nor to form any associations between views and vectors. However, there remains some experimental evidence not fully explained by this base model that may point towards the existence of a more complex or integrated mental map in insects.