BoxRemap remaps the cubical domain of a cosmological simulation into simple non-cubical shapes. It can be used for on-the-fly remappings of the simulation geometry and is volume-preserving; remapped geometry has the same volume V = L3 as the original simulation box. The remappings are structure-preserving (local neighboring structures are mapped to neighboring places) and one-to-one, with every particle/halo/galaxy/etc. appearing once and only once in the remapped volume.