Discontinuous precipitation (DP) and dissolution (DD) reactions represent a group of diffusional solid-solid phase transformations during which the solute redistribution accompanying the formation of a new phase occurs on a nanometer scale just at the moving reaction front. Such solute changes can be sufficiently revealed by means of analytical electron microscopy (AEM). A number of systematic investigations made using AEM during the last decade in DP and DD reactions is reviewed and critically analyzed. Measurements of the solute concentration profiles in the direction parallel to the reaction front are strongly emphasized, as they allow to verify the theoretical models of discontinuous reactions and to consider discontinuous reactions as local phenomena in the form of a number of events occurring in individual cells. It is shown that the use of the local approach to the kinetics of discontinuous precipitation and discontinuous dissolution reactions results in a substantial diminishing or even eliminating of the discrepancies existing between the values of the grain boundary diffusivities calculated from the equations of Cahn and Petermann-Hornbogen, using average or global values of the kinetic parameters. Simultaneously the obtained diffusivity corresponds well with the tracer diffusivity values obtained by the sectioning technique convincingly showing that there is no substantial difference between the rate of the diffusion process at moving and stationary grain boundaries. A discussion is also presented on how the results obtained via AEM can be used for an optimization of the occurrence of the discontinuous solid state reaction. Also the directions of the future work are indicated and unsolved problems are specified.