An hydrolysis stage of an anaerobic digestion (AD) process includes supercritical treatment in a reactor (R1). The treatment may use sub-stoichiometric oxygen so that there is not full oxidation. Effluent from the supercritical treatment may be used to pre-heat the organic matter in-feed to the supercritical treatment. There may be a second supercritical treatment in a second reactor (R2). This may have full stoichiometric oxygen, to oxidize the solids from the first reactor (R1) effluent. Furthermore, it provides heat for pre-heating its own in-feed and also that for the first reactor (R1). The output from the hydrolysis may be only clarified liquid, leading to particularly efficient downstream AD processes. The second supercritical treatment rector may oxidize also fed-back solids from the final stage of the AD.