In a fluid flow rate measuring apparatus having a sensor including an electric heater resistance wire and first and second temperature dependent resistance wires and a measuring circuit, each of the resistance wires is wound to be held firmly in place on a supporting member made of an electrical insulating material, and the electric heater resistance wire and the first temperature dependent resistance wire, which is responsive to the temperature of the fluid under measurement heated by the electric heater resistance wire, are wound on the same supporting member alternately and close to each other so that the winding direction thereof makes an angle from 60.degree. to 120.degree. to the direction of the flow of the fluid under measurement. Further, each of the supporting members is constructed and arranged so as to minimize the influence of the heat capacity thereof and also to avoid the occurrence of turbulence in the flow of the fluid which is detrimental to the flow rate measurement. This construction can provide a fluid flow rate measuring apparatus having high precision, improved response and elevated reliability.