A hydrodynamic bearing having an improved air discharging path for discharging air from the bearing when a shaft is inserted into the bearing, and an apparatus for driving a polygonal mirror using the hydrodynamic bearing. The hydrodynamic bearing includes: a bearing housing having a hollow portion filled with oil, and first and second stepped mounting recesses formed at one end of the hollow portion; a thrust plate installed in the first mounting recess so that one surface of the thrust plate is in contact with one end of the shaft, for supporting the shaft in a thrust direction, a diameter of the thrust plate being smaller than that of the first mounting recess: a first air discharging path formed at the thrust plate and/or the first mounting recess, for discharging air out of an inside of the bearing housing; a cap installed in the second mounting recess, for supporting a lower portion of the thrust plate; and a second air discharging path formed at the cap for communicating with the first air discharging path through a space between the thrust plate and the first mounting recess to discharge outwardly the air transferred from the first air discharging path.