A multi-stage centrifugal compressor comprising: a rotor including a rotor main body which extends along an axis extending in a horizontal direction, and impellers which are fixed to an outer surface of the rotor main body and which are provided in a plurality of stages in an axial direction; a plurality of diaphragms, each diaphragm including a guide flow path for guiding a fluid discharged radially outward from the impeller radially inward, and a communication hole which extends downward in a perpendicular direction from a bottom portion of the guide flow path; a casing which accommodates the plurality of diaphragms which are arranged in the axial direction corresponding to each of the plurality of stages of the impellers therein; and an axial flow path which extends in the axial direction so as to connect the plurality of the communication holes to each other, wherein the casing includes a suction nozzle which is provided on a first end portion side in the axial direction and guides a working fluid from the outside of the casing to the impeller on a first stage on the first end portion side, an ejection nozzle which is provided on a second end portion side in the axial direction and ejects the working fluid discharged from the impeller on the final stage on the second end portion side to the outside of the casing, and a drain flow path which is provided in the axial direction between the communication hole formed at a position that is the closest to the suction nozzle in the axial direction and the communication hole formed at a position that is the closest to the ejection nozzle in the axial direction, and causes the axial flow path and the outside of the casing to communicate with each other, wherein the axial flow path is formed between an outer circumferential surface of the plurality of diaphragms and an inner circumferential surface of the casing, characterized in that the inner circumferential surface of the casing is inclined so as to gradually lower down toward the drain flow path. a rotor including a rotor main body which extends along an axis extending in a horizontal direction, and impellers which are fixed to an outer surface of the rotor main body and which are provided in a plurality of stages in an axial direction; a plurality of diaphragms, each diaphragm including a guide flow path for guiding a fluid discharged radially outward from the impeller radially inward, and a communication hole which extends downward in a perpendicular direction from a bottom portion of the guide flow path; a casing which accommodates the plurality of diaphragms which are arranged in the axial direction corresponding to each of the plurality of stages of the impellers therein; and an axial flow path which extends in the axial direction so as to connect the plurality of the communication holes to each other, wherein the casing includes a suction nozzle which is provided on a first end portion side in the axial direction and guides a working fluid from the outside of the casing to the impeller on a first stage on the first end portion side, an ejection nozzle which is provided on a second end portion side in the axial direction and ejects the working fluid discharged from the impeller on the final stage on the second end portion side to the outside of the casing, and a drain flow path which is provided in the axial direction between the communication hole formed at a position that is the closest to the suction nozzle in the axial direction and the communication hole formed at a position that is the closest to the ejection nozzle in the axial direction, and causes the axial flow path and the outside of the casing to communicate with each other, wherein the axial flow path is formed between an outer circumferential surface of the plurality of diaphragms and an inner circumferential surface of the casing, characterized in that the inner circumferential surface of the casing is inclined so as to gradually lower down toward the drain flow path. an ejection nozzle which is provided on a second end portion side in the axial direction and ejects the working fluid discharged from the impeller on the final stage on the second end portion side to the outside of the casing, and a drain flow path which is provided in the axial direction between the communication hole formed at a position that is the closest to the suction nozzle in the axial direction and the communication hole formed at a position that is the closest to the ejection nozzle in the axial direction, and causes the axial flow path and the outside of the casing to communicate with each other, wherein the axial flow path is formed between an outer circumferential surface of the plurality of diaphragms and an inner circumferential surface of the casing, characterized in that the inner circumferential surface of the casing is inclined so as to gradually lower down toward the drain flow path.