Strong evidences supporting the occurrence of the nuclear transmutation reactions have been obtained by the electrolysis of light and heavy water mainly with gold and palladium electrodes. Steady excess energy evolution ranging from 0.2 to 1 W was observed during the electrolysis with gold electrode in neutral and alkaline light water solutions. At the same time, various unexpected elements, e.g. mercury, osmium, krypton, nickel, iron, silicon, magnesium, etc. were found to be produced on/in the gold electrode after the electrolysis. The amount of iron reached some 10 µg. In the case of palladium electrode electrolyzed in heavy water solution xenon, tin, cadmium, copper, iron, chromium, titanium etc. were detected. The isotopic distributions of above elements were evidently deviated from their natural isotopic abundance. In addition, some 100 µg, of fine porous precipitates were obtained when electrolyzed at a current density > 0.2 A/cm². The major component was gold, however several percents of mercury, osmium, hafnium, iron and silicon with anomalous isotopic compositions were contained. The appearance of the gold electrode surface after the electrolysis was very anomalous on which a lot of volcano-like micro craters were created. The structure of the outside wall was very porous, very alike to the structure of the precipitates. In the inside wall, there lied layers of Au (111)-like crystals. This shows the occurrence of the recrystallization of gold substrate, suggesting an extraordinary elevated heat evolution. It is considered from these results that some nuclear transmutation reaction occurred in that craters and the precipitates containing above elements were spewed out into the solution. In this review the details of these phenomena supporting the reality of the solid state nuclear transmutation will be described reinforcing several data of other authors.