Methods and apparatus for an oxy-fuel combustion power cycle are provided, including converting gaseous carbon dioxide to liquid and/or supercritical carbon dioxide which may include the use of a cryogenic pump, removing a portion of the liquid and/or supercritical carbon dioxide from the cycle, combusting oxygen and a combustion fuel with the remaining liquid and/or supercritical carbon dioxide in an oxy-fuel combustor to generate steam and additional liquid and/or supercritical carbon dioxide which replaces the portion of the liquid and/or supercritical carbon dioxide sequestered from the cycle.