A theory on the growth of island films from binary melts or a vapor phase under non-isothermal conditions is developed. A system in which the growth of island films takes place is thermally isolated, and both diffusion and temperature fields exist in the system. Temperature fields are formed due to evolution of the latent heat of crystallization or the heat of sublimation. The diffusion and temperature fields and self-consistent with each other. A full set of equations describing this process is derived. By solution of this set of equations, the rules governing the variation of the mean radius and the island density in time with different mechanisms for mass or heat transfer were found. It is shown that islands grow due to two processes, diffusional or thermal. One of these processes, namely, the diffusional one, is related to a variation in the supersaturation, while the thermal process occurs due to a change in the overcooling. It is also shown that the rate of island growth is determined by the factor that relates coefficients of the diffusion and thermal conductivity. Kinetic phase diagrams for binary systems are defined. A distribution relating the size of islands to their composition is obtained for the island films growing from binary systems, which gives rise to a continuous range of solid solutions in the solid state. The law of the variation of composition with time is found. Open systems are discussed in the second part of the work. The possibility of controlling the processes of growth of islands by means of sinks (sources) of heat or air is demonstrated. A model of growth of island film assemblies from melts having the eutectic composition is constructed. The possibility of controlling the structure or composition of continuous films is demonstrated.