Abstract: The invention provides an eccentric subsurface safety valve that may include an eccentric body member having a thick side, a thin side, and a longitudinal bore therethrough. A closure member, such as a curved flapper, is mounted within the body member to control fluid flow through the longitudinal bore, and is moveable between an open and a closed position. The curved flapper may be disposed within a recess in the thick side of the eccentric body member when in its open position. A valve actuator, such as a flow tube, may be disposed within the body member and remotely shiftable to move the curved flapper between its open and closed positions. The flow tube may be shifted downwardly in response to movement of a piston. At least one return spring and/or a contained volume of pressurized gas may be provided to urge the flow tube away from the curved flapper. The piston may be connected to an eccentric plate slidably disposed about the flow tube and between a first and a second shoulder on the flow tube.