A combustion system for operational connection to an energy conversion system and a method of providing thermal energy to the energy conversion system. The system comprises a combustor to be oxy-fired at above atmospheric pressure, using solid, liquid or gaseous fuels, with a supply of oxygen and supercritical carbon dioxide. The combustion gases from the combustor are delivered to a heat exchanger which interfaces with the energy conversion system. Temperatures in the combustor, and the delivery temperature to the heat exchanger, are controlled by selective recirculation of at least part of the combustion gases to the combustor, and by modulating the supply of oxygen and fuel to the combustor. Any combustion gases which are not recirculated are processed to separate carbon dioxide for use or sequestration. The system and method substantially eliminate emissions of carbon dioxide, while providing a highly efficient supply of thermal energy to the energy conversion system.