Procedural, technical and operational difficulties that arise with the use of flowmeters, both ultrasonic flowmeters and flowmeters constructed with the use of constriction devices (orifice-based flowmeters), for measuring feedwater flow rates at nuclear power plant reactors are considered. Results of measurements of these flowmeters are used to calculate the thermal power of the reactor plant and are important for controlling the nuclear reactor. It is shown that periodic inspection of welded-in orifice devices used at nuclear power plants of Russian design is not possible. Practical aspects of tests of ultrasonic flowmeters under normal conditions with assigned high precision for operation under the conditions of a power-generating unit are considered. New approaches to evaluation (inspection) of the state of constriction devices used in cooling agent flowmeters that are not accessible to examination throughout the entire operating period at nuclear power plants are proposed. It is shown that a method of direct comparison of the results of simultaneous measurements with a more precise mobile ultrasonic flowmeter may be used to estimate the error of flowmeters constructed with the use of constriction devices. A unified method is suggested for evaluating the state of constriction devices without dismantling the devices from the pipeline based on information on the total error of the flowmeters along with a complex of measures for the development of technical, procedural, and managerial means of controlling the reliability of flowmeter measurements with the use of constriction devices and ultrasonic and flowmeters.