A carbon dioxide compression system comprising: at least a first compressor stage (1, 3, 4) and a second compressor stage (2, 4, 5) arranged in series; at least one inter-stage heat exchanger (15, 30, 32, 42, 44, 46) arranged to receive a compressed gas containing carbon dioxide flowing from said first compressor stage into said second compressor stage and to remove waste heat therefrom, an energy conversion unit, wherein at least part of said waste heat is recovered and transformed into mechanical energy; wherein respective inter-stage heat exchangers (15, 30, 32, 42, 44, 46) are arranged between pairs of sequentially arranged compressor stages (1, 2; 3, 4; 4, 5); wherein a respective fluid path is provided for each said inter-stage heat exchangers to circulate said cooling fluid in said inter-stage heat exchangers and remove waste heat from compressed gas containing carbon dioxide circulating through said inter-stage heat exchangers; and wherein said fluid paths are in fluid communication with said energy conversion unit, such that said cooling fluid from said inter-stage heat exchangers is processed in said energy conversion unit to recover energy therefrom; characterized by a pumping section wherein the compressed gas containing carbon dioxide exiting the last one of the serially arranged compressor stages is further cooled and finally pumped to the final desired pressure, the pumping section comprising a further heat exchanger arranged to receive compressed gas containing carbon dioxide flowing from a last one of said serially arranged compressor stages, to remove waste heat therefrom; a pump arranged to receive compressed gas containing carbon dioxide from said further heat exchanger and deliver gas containing carbon dioxide in a delivery duct; and a further fluid path provided to circulate a cooling fluid through said further heat exchanger to remove waste heat from the compressed gas containing carbon dioxide between said last compressor stage and said pump, said further fluid path being arranged to deliver said cooling fluid exiting said further heat exchanger to said energy conversion unit. at least a first compressor stage (1, 3, 4) and a second compressor stage (2, 4, 5) arranged in series; at least one inter-stage heat exchanger (15, 30, 32, 42, 44, 46) arranged to receive a compressed gas containing carbon dioxide flowing from said first compressor stage into said second compressor stage and to remove waste heat therefrom, an energy conversion unit, wherein at least part of said waste heat is recovered and transformed into mechanical energy; wherein respective inter-stage heat exchangers (15, 30, 32, 42, 44, 46) are arranged between pairs of sequentially arranged compressor stages (1, 2; 3, 4; 4, 5); wherein a respective fluid path is provided for each said inter-stage heat exchangers to circulate said cooling fluid in said inter-stage heat exchangers and remove waste heat from compressed gas containing carbon dioxide circulating through said inter-stage heat exchangers; and wherein said fluid paths are in fluid communication with said energy conversion unit, such that said cooling fluid from said inter-stage heat exchangers is processed in said energy conversion unit to recover energy therefrom; characterized by a pumping section wherein the compressed gas containing carbon dioxide exiting the last one of the serially arranged compressor stages is further cooled and finally pumped to the final desired pressure, the pumping section comprising a further heat exchanger arranged to receive compressed gas containing carbon dioxide flowing from a last one of said serially arranged compressor stages, to remove waste heat therefrom; a pump arranged to receive compressed gas containing carbon dioxide from said further heat exchanger and deliver gas containing carbon dioxide in a delivery duct; and a further fluid path provided to circulate a cooling fluid through said further heat exchanger to remove waste heat from the compressed gas containing carbon dioxide between said last compressor stage and said pump, said further fluid path being arranged to deliver said cooling fluid exiting said further heat exchanger to said energy conversion unit.