High performance, low emission engines, multiple cylinder engines and operating methods based on compression ignition of a combustion chamber charge. In accordance with the methods, fuel is injected into the combustion chamber after the exhaust valve closes, and air is injected into the combustion cylinder before the end of the compression stroke in an amount to limit the temperature after ignition to less than the temperature at which NOX forms. After ignition and during the power stroke, more air is injected into the combustion chamber to sustain combustion until all fuel is consumed, the air being injected so as to again limit the combustion temperatures to less than the temperature at which NOX forms. The air injected during the compression stroke is injected at substantially the same combustion chamber pressure as the air injected during the power stroke, so that air may efficiently be injected from the same pressurized source. The fuel injected into the combustion chamber may be all the fuel needed, or additional fuel may be injected during the power stoke.