The general relativistic magnetohydrodynamical approximation is used to study plasma flows near rotating black holes and the formation of jets. The magnetized jet from the rotating black hole is found to be efficiently accelerated to Lorentz factors of Γ ≳ 10. The magnetized jet is collimated by the corona, which has a wind component that reaches Γ ≲ 3. The rotation within the jet keeps it marginally stable to current‐driven instabilities at large radii. These instabilities limit the efficiency of the magnetic acceleration and collimation, while providing a new pathway for converting magnetic to thermal energy that itself can lead to acceleration.