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Work on the redefinition of the kilogram and mole underway in many countries with experiments employing watt balances is analyzed. It is found that at present the reasons for the discrepancies in measurements of the Avogadro and Planck constants by different methods are not clear and that the required measurement accuracy has not been attained. Thus, studies with watt balances and crystalline silicon spheres must be continued in the framework of new projects.

Selection of a fundamental physical constant for the purpose of redefinition of the updated SI base measurement unit according to the principle of concordance of dimensions of the constant and of the unit of measurement, when they either coincide or differ by a factor having the dimension of time, is proposed. A quantitative criterion for reconciling the new prototypes of the kilogram with its international prototype and a quantitative limitation of their instability are proposed. The possibility of preserving a relation between the definitions of the mole and the kilogram in the updated SI is considered. It is proposed that the kilogram and mole be redefined using a fixed value of the Avogadro constant, the kelvin, and the ampere, after, that is, the measurement accuracy of the Boltzmann constant has been improved and after closure of the “quantum triangle.”

Some trends in research on the physics underlying metrology as the science of measurement are examined. The basic problems and the prospects for their solution are analyzed.

The paper discusses possible signs of time variation in the constants for the extended standard model. Existing experimental constraints on the magnitude of the changes are given in order to allow for the possible effects on the metrological characteristics of primary standards for the units of physical quantities.

Some methods of determining the constants of the Standard Model, the values of the constants established at the present time, and the dependence of the coupling constants of the strong, electromagnetic and weak interactions on the characteristic energy of the process are presented. Possible generalizations of the Standard Model, which lead to a reduction in the number of fundamental constants, are considered.

Fundamental questions of contemporary metrology connected with redefinition of the basic units of physical quantities are discussed. Some new definitions of the kilogram, ampere, Kelvin, and mole are suggested and their consequences are analysed. The possibility of fixing some fundamental physical constants at the level of accuracy contemporary for 2011 is considered. Alongside the practical system of units, use of a fundamental (theoretical) system is proposed. The importance is noted of considering possible variations of physical constants for a fundamental system of units, whereas in practical metrology these variations are not currently essential.

The main versions of the redefinition for the international prototype kilogram are examined for best accuracy and transmitting the mass unit. It is concluded that it is most promising to use Avogadro’s number and the atomic mass unit in order to transfer to a new standard for the kilogram, i.e., to introduce an atomic kilogram.

Contemporary methods are presented for redefining the unit of mass standard, i.e., the kilogram prototype. The main tendencies in improving these methods in order to reduce the uncertainty of reproducing the unit of mass are discussed. It is noted that the use of precise values of a number of fundamental physical constants, primarily the Avogadro number, is important.

Contemporary methods are considered for storage and reproduction of the dimensions of basic SI units, i.e., length, time, and mass. The main trends are discussed for improving these methods in order to reduce reproduction errors for the dimensions of units. The importance of using results of theoretical calculations for microprocess characteristics and precise values for a number of fundamental physical constants is noted.

The paper discusses the fundamental physical constants, the classification of them, the number of them, the definition accuracy, and the relationships to fundamental physical theories (interactions). Basic concepts are presented on gravitational relativistic metrology related to precision space-time measurements. The main attention is given to Newton’s gravitational constant G as regards the measurement of the absolute value and the possible variations in time and space as predicted by interaction union theories. Principles are presented for a proposed space experiment to measure G and its variations, whose implementation will enable one to improve on current data from appropriate laboratory experiments on Earth by 2–3 orders of magnitude.