As a result of the reassessment of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory for the kinetics of nucleation and growth transformations, an integral-equation formulation has been developed instead of the well-known and widely used Avrami equation. The presented formulation considers interfacial and diffusional growths, in one, two, and three dimensions, with both time-dependent and time-invariant nucleation and growth rates. The integral-equation model corrects reported inadequacies of the KJMA theory when applied in numerous experiments and various solid-state transformations. It is shown that in the example cases examined in this paper, crystallization from the amorphous state in melt-spun ribbons, isothermal aging of CuAlZn, pearlitic transition in an eutectoid steel, and crystallization in a PEKK polymer, the thermodynamic and kinetic interpretation and parameters extracted from best fits of the Avrami equations to the experimental data are erroneous. The KJMA formulation is a simplification of the real physical conditions. The main limitation of the new model is that almost all the integral equations representing the kinetics of solid-state transformations have no analytical solutions.