A steam cleaning apparatus includes water and wax containers to supply water and wax, a pump to discharge a mixture of water and wax, a combustion chamber to receive the discharged mixture in an internal pipe and to spray the mixture through a nozzle such that the mixture can be discharged along the pipe, a burner to heat the pipe of the combustion chamber with a fuel from a fuel container to convert the mixture into steam having high temperature and high pressure, a steam gun to discharge the high temperature and pressure steam from the combustion chamber to an outside, and a controller to control the water container, wax container, pump, combustion chamber, fuel container, and burner and to allow the mixture to be discharged to the outside after being converted to the high temperature and pressure steam. A method of driving the apparatus is also provided.