Publisher Summary The dissolution of a sample is an integral part of most procedures and is mandatory if separation/preconcentration techniques are used. The three basic groups of decomposition methods include acid attack, fusion and decomposition in gas streams. They are applied to powder samples (finer than 100 mesh) for the sake of speed. Attack with strong inorganic acids (HCl, HNO3, HCl04, and H2SO4) and others (H3PO4 and HF) is a common method for the dissolution of metals, alloys, geochemical samples, and organic materials. The fusion process involves a heterogeneous reaction, the rate of which is limited by the contact surface, collection of insoluble precipitate on the surface and diffusion conditions. The decomposition methods reviewed in the chapter undergo continuous evolution mostly in terms of vessel design and heat transfer that aims at improving the speed and completeness of the process. Dissolution techniques include open vessel dissolution, pressure bomb decomposition, decomposition with acid vapours, and microwave-assisted digestion.