A rotary machining apparatus comprising: a support body that has a surface intersecting a predetermined axis and supports a workpiece; a shaft-equipped spherical seat that is installed to face the surface of the support body and performs swinging movement with respect to the workpiece; characterized by a cover (21, 21a, 21b), for appropriate environmental protection against the swinging movement, that is installed to be able to rotate around the predetermined axis and relatively rotates around the predetermined axis with respect to the shaft-equipped spherical seat in response to swinging movement of the shaft-equipped spherical seat, wherein: either the cover (21, 21a, 21b) has an asymmetrical circumferential wall (53, 53a) around the predetermined axis, and a height of the circumferential wall (53, 53a) varies depending on position in the circumferential direction around the predetermined axis; or the rotary machining apparatus further comprises a guide ring that is disposed around the machining subject as the workpiece, the cover is constituted of a flexible material and/or is constituted to be partially extensible and contractible in axial dimension in the circumferential direction, and the one axial end portion of the cover is supported or placed on the guide ring to be able to rotate about the reference axis and the axially opposite end portion of the cover is supported by a part of the shaft-equipped spherical seat on the one axial end side to be able to rotate about the central axis of the shaft-equipped spherical seat. a support body that has a surface intersecting a predetermined axis and supports a workpiece; a shaft-equipped spherical seat that is installed to face the surface of the support body and performs swinging movement with respect to the workpiece; characterized by a cover (21, 21a, 21b), for appropriate environmental protection against the swinging movement, that is installed to be able to rotate around the predetermined axis and relatively rotates around the predetermined axis with respect to the shaft-equipped spherical seat in response to swinging movement of the shaft-equipped spherical seat, wherein: either the cover (21, 21a, 21b) has an asymmetrical circumferential wall (53, 53a) around the predetermined axis, and a height of the circumferential wall (53, 53a) varies depending on position in the circumferential direction around the predetermined axis; or the rotary machining apparatus further comprises a guide ring that is disposed around the machining subject as the workpiece, the cover is constituted of a flexible material and/or is constituted to be partially extensible and contractible in axial dimension in the circumferential direction, and the one axial end portion of the cover is supported or placed on the guide ring to be able to rotate about the reference axis and the axially opposite end portion of the cover is supported by a part of the shaft-equipped spherical seat on the one axial end side to be able to rotate about the central axis of the shaft-equipped spherical seat. either the cover (21, 21a, 21b) has an asymmetrical circumferential wall (53, 53a) around the predetermined axis, and a height of the circumferential wall (53, 53a) varies depending on position in the circumferential direction around the predetermined axis; or the rotary machining apparatus further comprises a guide ring that is disposed around the machining subject as the workpiece, the cover is constituted of a flexible material and/or is constituted to be partially extensible and contractible in axial dimension in the circumferential direction, and the one axial end portion of the cover is supported or placed on the guide ring to be able to rotate about the reference axis and the axially opposite end portion of the cover is supported by a part of the shaft-equipped spherical seat on the one axial end side to be able to rotate about the central axis of the shaft-equipped spherical seat.