A critical analysis of the nucleation theorem in its different formulations is presented. A new formulation of the nucleation theorem, mathematically widely equivalent to the form given by Oxtoby and Kashchiev in 1994, is developed. This formulation is, however, more easily applicable to the interpretation of experimental results as compared with the original expressions given by Kashchiev and Oxtoby. It can be utilized straightforwardly also in cases where the original version cannot be employed and allows, in addition, a variety of further theoretical developments. It is shown, moreover, first in the framework of Gibbs's theory of heterogeneous systems that the nucleation theorem holds not only for critical clusters but for clusters of arbitrary sizes as well. The new formulation of the nucleation theorem is applied then to the analysis of different cases of phase formation. It is demonstrated, as a first application, that the original formulation of the nucleation theorem by D. Kashchiev (J. Chem. Phys. 76, 5098 (1982)) can be extended to multicomponent systems as well. The conditions of validity of this particular formulation are discussed both for the original version (developed for one-component systems) and the generalization to multicomponent systems considered here. It turns out that the original formulation of the nucleation theorem is strictly valid only for incompressible cluster phases (i.e., for systems, strictly speaking, nonexistent in nature). Approximately, it may be employed for real systems as well but only in the limit of moderate supersaturations. As additional illustrations, the new general formulation of the nucleation theorem is applied then to different special cases of interest, analyzed and not analyzed before. A comparison with previous studies on similar topics is made. It is shown further that an alternative thermodynamic derivation of the nucleation theorem both for clusters of critical as well as for arbitrary sizes can be given at certain specified conditions based on the van der Waals approach to the description of heterogeneous systems. Finally, the limits of validity of the nucleation theorem are analyzed with respect to its applicability to real systems; i.e., the problem is discussed to what extent the nucleation theorem may give an adequate description of properties of critical clusters determining the nucleation process in real systems or in computer models of nucleation phenomena.