Exact results for the first passage time and leapover statistics of symmetric and one-sided Lévy flights (LFs) are derived. LFs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one-sided LFs and, surprisingly, with an index alpha/2 for symmetric LFs. The first passage time distribution scales like a power law with an index 1/2 as required by the Sparre-Andersen theorem for symmetric LFs, whereas one-sided LFs have a narrow distribution of first passage times. The exact analytic results are confirmed by extensive simulations.