A fan comprising an electrically driven motor having a stator and having a rotor which is mounted so as to be rotatable about an axis of rotation, wherein a radial and an axial direction (RR, AR) of the fan are defined on the basis of the axis of rotation, and wherein the motor has at least one winding through which an electrical current flows during operation; a fan impeller which is fastened rotationally conjointly to the rotor and which serves for drawing in and conveying a gaseous medium; a cooling body having an inner wall which delimits an interior space for accommodating the motor, and having air-guiding elements which extend in each case in the axial direction over a major part of the longitudinal extent of the winding, through which electrical current flows, and in each case rectilinearly over the entire longitudinal extent of the stator, in order to conduct the gaseous medium, which is conveyed by the fan impeller, along the cooling body for motor cooling purposes; a housing having an outer wall, which has a cylindrical inner surface, and which delimits a cavity for accommodating the cooling body and the motor; and a diffuser, which comprises guide blades and is provided along the axial direction between the fan impeller and the cooling body, in order to divert the gaseous medium, which is conveyed by the fan impeller, into the axial direction toward the cooling body; wherein each of the air-guiding elements of the cooling body is assigned a guide blade, and wherein the air-guiding elements in each case adjoin the guide blades in the axial direction and transition into said guide blades in a flush manner, and wherein the inner wall of the cooling body has an outer surface whose diameter decreases continuously in the axial direction with increasing distance from the fan impeller along substantially the entire longitudinal extent of the inner wall. an electrically driven motor having a stator and having a rotor which is mounted so as to be rotatable about an axis of rotation, wherein a radial and an axial direction (RR, AR) of the fan are defined on the basis of the axis of rotation, and wherein the motor has at least one winding through which an electrical current flows during operation; a fan impeller which is fastened rotationally conjointly to the rotor and which serves for drawing in and conveying a gaseous medium; a cooling body having an inner wall which delimits an interior space for accommodating the motor, and having air-guiding elements which extend in each case in the axial direction over a major part of the longitudinal extent of the winding, through which electrical current flows, and in each case rectilinearly over the entire longitudinal extent of the stator, in order to conduct the gaseous medium, which is conveyed by the fan impeller, along the cooling body for motor cooling purposes; a housing having an outer wall, which has a cylindrical inner surface, and which delimits a cavity for accommodating the cooling body and the motor; and a diffuser, which comprises guide blades and is provided along the axial direction between the fan impeller and the cooling body, in order to divert the gaseous medium, which is conveyed by the fan impeller, into the axial direction toward the cooling body; wherein each of the air-guiding elements of the cooling body is assigned a guide blade, and wherein the air-guiding elements in each case adjoin the guide blades in the axial direction and transition into said guide blades in a flush manner, and wherein the inner wall of the cooling body has an outer surface whose diameter decreases continuously in the axial direction with increasing distance from the fan impeller along substantially the entire longitudinal extent of the inner wall.