Metrics admitting projective and c-projective vector fields

Vector fields on a smooth manifold with an affine connection are called projective (resp. c-projective) if their local flows preserve geodesics (resp. J-planar curves). In this expository paper, metrics with such vector fields are discussed. Emphasis is put on Lie’s classical 1882 problem of finding a local description of surfaces with projective vector fields, known results on this problem, as well as on various extensions in the literature.