A torque fluctuation absorbing apparatus, comprising: a center plate disposed rotatably; a side plate assembly (17, 18) disposed coaxially with and rotatably relative to the center plate; an elastic member(s) (34, 35) housed within a space delimited by the side plate assembly (17, 18) so as to absorb torque fluctuations produced between the side plate assembly (17, 18) and the center plate; and a cover member secured to the center plate in intimate contact therewith in a continuous manner throughout in a circumferential direction, the cover member being slidably pressure contacted with the side plate assembly (17, 18) in a continuous manner throughout in the circumferential direction, in such a manner that a space housing the elastic member(s) (34, 35) therein will be shielded from outside, wherein the cover member is an annular plate-shaped member, a surface of the cover member attached to the center plate is pressure contacted with the side plate assembly (17, 18) such that performance of the intimate contact of the cover member with respect to the side plate assembly (17, 18) is elevated when an atmospheric pressure within a space housing the elastic member(s) (34, 35) therein has become lower than an outside atmospheric pressure, and a side of the side plate assembly (17, 18) directed to the center plate has an inner peripheral end face radially outwardly of a portion of the cover member at which the cover member is secured to the center plate, a corner of the inner peripheral end face of the side plate assembly (17, 18) remoter from the center plate including an abutting portion disposed so as to have pressure contact with the cover member. a center plate disposed rotatably; a side plate assembly (17, 18) disposed coaxially with and rotatably relative to the center plate; an elastic member(s) (34, 35) housed within a space delimited by the side plate assembly (17, 18) so as to absorb torque fluctuations produced between the side plate assembly (17, 18) and the center plate; and a cover member secured to the center plate in intimate contact therewith in a continuous manner throughout in a circumferential direction, the cover member being slidably pressure contacted with the side plate assembly (17, 18) in a continuous manner throughout in the circumferential direction, in such a manner that a space housing the elastic member(s) (34, 35) therein will be shielded from outside, wherein the cover member is an annular plate-shaped member, a surface of the cover member attached to the center plate is pressure contacted with the side plate assembly (17, 18) such that performance of the intimate contact of the cover member with respect to the side plate assembly (17, 18) is elevated when an atmospheric pressure within a space housing the elastic member(s) (34, 35) therein has become lower than an outside atmospheric pressure, and a side of the side plate assembly (17, 18) directed to the center plate has an inner peripheral end face radially outwardly of a portion of the cover member at which the cover member is secured to the center plate, a corner of the inner peripheral end face of the side plate assembly (17, 18) remoter from the center plate including an abutting portion disposed so as to have pressure contact with the cover member.