Abstract: The motion of a charged particle in a magnetic field has become a popular object of study in mathematical physics, dynamical systems, symplectic and differential geometry. In this talk, magnetic billiards will be interpreted as billiards in Finsler geometry; this interpretation has geometric consequences, such as the existence of magnetic "whispering galleries". I will also characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. This is a magnetic analog of Hilbert's Fourth problem asking to describe the Finsler metrics whose geodesics are straight lines. Both problems have solutions in terms of special symplectic structures on the spaces of extremals. The magnetic analog of Hilbert's fourth problem is closely related to another well studied subject, the Pompeiu problem.