NOVELTY - A digester for the digestion of organic waste or organic feedstock, comprises several of at least three reaction chambers, where the organic waste or feedstock passes through each reaction chamber before exiting the digester; and one or more of a collection tank, pH sensor, chemical feed tank, microbe feed tank, cleaning apparatus, removing device, temperature control devices, gas capture chamber, nutrient tank, recycle chamber, insulation material, waste concentration device, waste diluting device, microbe product chambers, anaerobic microbe recycle chambers, and partitioning devices.    USE - The invention is used for digestion of organic waste or organic feedstock (claimed). It is used for converting predominantly organic waste materials, e.g. sludge or animal waste, into useful byproducts. It is also used to produce methane and/or hydrogen gas (claimed). It may also be used to manufacture pharmaceutical products from certain organic chemical feedstocks.    ADVANTAGE - The inventive digester reduces predominantly organic waste materials, e.g. wastewater treatment plant sludge or animal wastes, into useful byproducts. It provides a series of environments that select for anaerobic microorganisms that efficiently digest wastewater sludge. By using naturally occurring microorganisms, and by compartmentalizing the selection of organisms that most effectively thrive on the material found in that particular compartment, the inventive digester efficiently and rapidly digests the waste. This efficiency is surprising because no bacteria normally need be added to the sludge, and the contents of the reactor need not be mechanically stirred. The overall size of the digester can be adjusted to handle a wide range of waste volumes. The inventive digester enables digestion to proceed at a high rate, reducing the residence time necessary to produce a clean effluent. It requires neither pre-digestion nor spiking with bacteria to initiate anaerobic digestion, only an additional carbon source may be needed, depending on the nature of the waste stream. Thus, if the waste source has a low microbe concentration, e.g. some industrial waste, the addition of some microorganisms may be helpful. Finally, the inventive digester is simple and energy-efficient since no internal moving parts are required.    DETAILED DESCRIPTION - A digester for the digestion of organic waste or organic feedstock, comprises several (n) of 3 reaction chambers, where the organic waste or feedstock passes through each reaction chamber before exiting the digester. Each reaction chamber comprises an inlet and an outlet. It is adapted to foster an anaerobic microbial digestion of organic waste or feedstock within the reaction chamber. The first reaction chamber is adapted to receive an influent stream of organic waste or feedstock through the inlet of the first reaction chamber, and to transfer partially digested organic waste or feedstock through the outlet of the first reaction chamber to the inlet of the second reaction chamber. The j-th reaction chamber is adapted to receive, through the inlet of the j-th reaction chamber, at least partially digested organic waste or feedstock from the outlet of the (j-1)-st reaction chamber. It is adapted to transfer partially digested organic waste or feedstock from the outlet of the j-th reaction chamber to the inlet of the (j+1)-st reaction chamber, where j is an integer so that j is greater than 1 and less than n. The n-th reaction chamber is adapted to receive, through the inlet of the n-th reaction chamber, partially digested organic waste or feedstock from the outlet of the (n-1)-st reaction chamber. It is adapted to transfer partially digested organic waste or feedstock through the outlet of the n-th reaction chamber.    The digester further comprises one or more of:    (i) a collection tank, positioned upstream of the reaction chambers, in which organic waste materials or feedstocks are collected prior to their introduction into the first reaction chamber;    (ii) a pH sensor positioned in communication with the contents of the reaction chambers or an inflow stream;    (iii) a chemical feed tank positioned in fluid communication with the reaction chambers that introduces desired chemicals;    (iv) a microbe feed tank positioned in communication with the reaction chambers that introduces selected microbes;    (v) a cleaning apparatus positioned in fluid connection with the reaction chambers;    (vi) a removing device for removing inorganic solids, positioned upstream of the first reaction chamber;    (vii) one or more temperature control devices controlling the temperature of the organic waste or feedstock, and positioned in fluid communication with the reaction chambers;    (viii) a gas capture chamber positioned in gaseous communication with one of the reaction chambers;    (ix) a nutrient tank in fluid communication with the reaction chambers;    (x) a recycle chamber positioned in fluid communication with the reaction chambers;    (xi) an insulation material positioned to insulate the contents of the reaction chambers;    (xii) a waste concentration device positioned in communication with the contents of the reaction chambers or the inflow stream;    (xiii) a waste diluting device positioned in communication with the contents of the reaction chambers or the inflow stream;    (xiv) one or more microbe product chambers positioned in communication with the reaction chambers that contain the selected microbes that further digest partially digested waste or feedstock to eventually form desired new products;    (xv) one or more anaerobic microbe recycle chambers positioned in communication with the reaction chambers that contain the selected microbes that further digest partially digested waste or feedstock for purpose of speeding start up or maintenance of the digesters; and    (xvi) one or more partitioning devices, e.g. partition(s), valve or baffle, used to partition the gas collected from the chambers.    INDEPENDENT CLAIMS are also included for:    (1) anaerobic digestion of organic waste or organic feedstock; and    (2) producing methane gas or hydrogen gas from organic waste or organic feedstock.    DESCRIPTION OF DRAWING(S) - The figure is a flow diagram of a process and a digester for anaerobically digesting organic wastes to reduce solids and produce a useful product, e.g. methane.