The dynamic evolution of the cluster-size distribution (CSD) in the precoalescence stages of nucleation and growth for sizes within and beyond the nucleation barrier layer (NBL) has been obtained. The existence of several universal kinetic laws of nucleation is predicted. In the precoalescence stages of nucleation, the CSD for sizes within and beyond the NBL obeys dynamic scaling relations. In a range of sizes beyond the NBL, the CSD exhibits a universal power law behavior, the exponent of which depends only on the interphase monomer transfer mechanism. Based on the results for the CSD, a general foundation is developed for nucleation kinetic measurements including the determination of the nucleation barrier and the interfacial monomer transport rate by measuring the CSD. A preliminary comparison with an experimental study confirms the predicted dynamic scaling and the power law behavior of the CSD in the early stages of nucleation and growth.