Publisher Summary The ability of the supersaturated old phase to remain a certain time in the state of metastable thermodynamic equilibrium (TDE) is one of the most remarkable features of the phase transitions of the first order and is because of the fact that the metastable state is separated from the truly stable one by an energy barrier. Besides, a differently high barrier corresponds to each concrete path along which the transition may proceed, and it might be expected that the actual path of the process is the one requiring the lowest energy expenses. The chapter discusses the two commonly used approaches for describing the tiny precursors of the new phase: the cluster approach and the density-functional approach. In the cluster approach, the nanoscopically small formation of the new phase is considered as a cluster of a certain number of molecules in it. The cluster itself is regarded as separated from the old phase by a phase boundary, and that makes it possible to say which of all molecules in the system are still in the old phase and which of them already belong to the new phase.