An engagement device includes: a first member including a plurality of engaged teeth; a second member arranged coaxially with the first member, the second member including a plurality of engaging teeth; a rotation unit configured to relatively rotate the first member and the second member about an axis; a movement unit configured to relatively move the first member and the second member in an axial direction; and a control unit configured to control operations of the rotation unit and the movement unit, and the engaged teeth are formed on the first member at a side opposing the second member along a circumferential direction around the axis, and the engaging teeth are formed on the second member at a side opposing the first member along the circumferential direction around the axis.