Major recent advances in the theoretical study of heterogeneous nucleation on macroscopic wettable centers of different nature are reviewed in the context of the classical scheme which uses the thermodynamics of a new phase nucleation to calculate the key kinetic characteristics of nucleation. The review centers on the kinetics of heterogeneous nucleation under conditions where a metastable state of the initial phase gradually forms - a situation in which the factors supporting the phase transition to the metastable state and then deepening the phase into the metastability region also remain active after the intense phase transition has begun. The formulation and control of the conditions of consistency for applying all the thermodynamic and kinetic elements of the theory are emphasized. The thermodynamics of interfaces are discussed in detail and a careful kinetic analysis is performed for the stage at which the main number of stably growing nuclei of the new phase is formed around wettable nuclei consisting of soluble or insoluble, surface-inactive or surface-active materials. The features common to barrier and barrierless heterogeneous nucleation processes are discussed, which open new possibilities in the experimental study of the phenomenon and offer new practical applications of the theory.