TECHNOLOGY FOCUS - ENVIRONMENT - Preferred Method: In the method (M1), the anaerobic water purification utilizes biomass carriers for supporting anaerobic microorganisms. The methods (M1) and (M2) further comprise collecting gas produced by the anaerobic water purification in a headspace. In the methods (M1) and (M2), the aerobic water purification includes moving bed biofilm reactor functionality or moving bed clarifying reactor functionality. In the method (M2), pressure created by the accumulation of the gas produced by the anaerobic water purification is operative to pump the anaerobic-treated water output from the anaerobic water purification to the aerobic water purification. The method (M2) further comprises supplying gas to the anaerobic water purification subsystem for causing relative movement of the biomass carriers, where supplying gas utilizes gas received from the headspace. In the anaerobic/aerobic liquid purification method, at least one of the anaerobic purifying and the aerobic purifying utilizes biomass carriers for supporting microorganisms. In the anaerobic/aerobic liquid purification method, at least one of the aerobic liquid purification subsystem and the anaerobic liquid purification subsystem includes moving bed biofilm reactor functionality. The anaerobic/aerobic liquid purification method further comprises supplying gas to the anaerobic liquid purification subsystem for causing relative movement of the biomass carriers, where supplying gas utilizes gas produced in the anaerobic liquid purification subsystem.    TECHNOLOGY FOCUS - INSTRUMENTATION AND TESTING - Preferred Components: The anaerobic water purification system further comprises: a gas collection volume located above the anaerobic water purification unit for collecting gas produced by the anaerobic water purification unit, where the gas collection volume is located in a headspace above the anaerobic water purification unit; and gas supply functionality for supplying gas to the anaerobic water purification unit for causing relative movement of the biomass carriers, where the gas supply functionality supplies gas received from the gas collection volume. The anaerobic water purification unit receives water to be treated at a location near its bottom or top. In anaerobic/aerobic water purification systems (A1) and (A2), the aerobic water purification subsystem is located physically above the anaerobic water purification subsystem. In anaerobic/aerobic water purification system (A1), the anaerobic water purification subsystem includes biomass carriers for supporting anaerobic microorganisms. The anaerobic/aerobic water purification systems (A1) and (A2) further comprise: a gas collection volume located above the anaerobic water purification subsystem and below the aerobic water purification subsystem for collecting gas produced by the anaerobic water purification subsystem, where the gas collection volume is located in a headspace above the anaerobic water purification subsystem; and gas supply functionality for supplying gas to the anaerobic water purification subsystem for causing relative movement of the biomass carriers, where the gas supply functionality supplies gas received from the gas collection volume. In the anaerobic/aerobic water purification systems (A1), (A2) and (A3), pressure created by the accumulation of the gas produced by the anaerobic water purification subsystem is operative to pump the anaerobic-treated water output from the anaerobic water purification subsystem to the aerobic water purification subsystem. In the anaerobic/aerobic water purification system (A1), the aerobic water purification subsystem includes moving bed biofilm reactor functionality. The anaerobic/aerobic water purification systems (A1) and (A3), the anaerobic water purification subsystem receives water to be treated at a location near its bottom. In the anaerobic/aerobic water purification systems (A2) and (A3), the aerobic water purification subsystem includes moving bed biofilm reactor functionality or moving bed clarifying reactor functionality. In anaerobic/aerobic water purification system (A2), the anaerobic water purification subsystem receives water to be treated at a location near its bottom or top. The anaerobic/aerobic water purification system (A3), the gas collection volume is located in a headspace above the anaerobic water purification subsystem. The anaerobic/aerobic water purification system (A3) further comprises gas supply functionality for supplying gas to the anaerobic water purification subsystem for causing relative movement of the biomass carriers. In the anaerobic/aerobic liquid purification system, pressure in the anaerobic liquid purification subsystem is operative to pump the anaerobic-treated liquid output from the anaerobic liquid purification subsystem to the aerobic liquid purification subsystem. The anaerobic/aerobic liquid purification system further comprises: a gas collection volume located in a headspace above the anaerobic liquid purification subsystem for collecting gas produced by the anaerobic liquid purification system; a gas supply mechanism for supplying gas to the anaerobic liquid purification subsystem, where the gas supply mechanism is connected to the gas collection volume; and liquid recirculation functionality. The aerobic liquid purification subsystem is located above the anaerobic liquid purification subsystem, where the anaerobic liquid purification subsystem includes biomass carriers for supporting anaerobic microorganisms. The aerobic liquid purification subsystem further comprises moving bed biofilm reactor functionality.