Maps that take lines to conics

We will discuss generalizations of the classical theorem of Moebius (1827): a one-to-one self-map of a real projective space that takes all lines to lines is a projective transformation. E.g. we study sufficiently smooth local maps taking line segments to parts of conics. A description of local maps taking line segments to circle arcs depends non-trivially on the dimension (the descriptioninvolves classical geometries, quaternionic Hopf fibrations, representations of Clifford algebras). For most dimensions, it is still missing.