The role of the fundamental constants and of measurement data on the Planck and Avogadro constants, the kelvin, and the electrical charge in the planned transition to new definitions of the four SI base units (kilogram, mole, ampere, and kelvin) based on fixed values of these constants is discussed. Keywords: redefinition of the SI base units, fundamental physical constants, dimension of a physical quantity, instability of the international prototype kilogram.There are three main unsolved problems in modern physics: determination of the four known physical interactions, including gravitation; dark energy and dark matter in the contemporary expansion of the universe; and possible variations in the fundamental physical constants, the fine structure constant, gravitation constant, ratio of the proton and electron masses, etc.In physical theories, constants characterize the stability properties of different forms of matter, objects, processes, and so on, and remain invariant under different circumstances, at least in terms of existing measurement uncertainties. Since the fundamental physical constants or combinations of them create natural scales for the basic units of measurements, they are referred to as fundamental. As science develops, physical theories are replaced by more general theories with their own sets of constants, with new relationships among them. Thus, we can only discuss a set of constants corresponding to the modern situation in the physical sciences, rather than some absolute set of fundamental physical constants [1].At present, the standard model of strong and electroweak interactions is consistent with most of the available experimental data. A fairly well tested (mainly in the solar system) theory of gravitation (Einstein's general theory) also exists. The standard cosmological model (the spatially flat Fridman model with a cosmological constant, ΛCDM) and its many modifications and generalizations play an important role in cosmology with its rapidly developing observational basis and theoretical interpretations. In each of these theories, there are problems with the corresponding set of constants [1,2], and with the further development of physics the existing set of fundamental physical constants (FPC) will probably change, but it is not currently known when this will happen.The systems of units of physical quantities are a part of the necessary toolkit of science. Like any toolkit, they must correspond to the current state of development of science. The redefinition of the Systeme Internationale (SI) base units now being prepared can be regarded as an answer to this need [3]. The planned reform of the SI relies on a proposal to define the base units by setting exact values of the corresponding FPC in accordance with the principle used in 1983 for redefining the meter.Since 2005, it has been proposed for the SI reform that, in particular, the values of the Planck h, Boltzmann k, and Avogadro N A constants and of the electronic charge e be fixed with zero uncertainty, and that this should serve...