Provided is a continuous method and apparatus of purifying carbon nanotubes. The continuous method and apparatus of purifying carbon nanotubes is characterized in a first purifying step for injecting a carbon nanotube liquid mixture containing an oxidizer into a purifying reactor under a sub-critical water or supercritical water condition at a pressure of 50 to 400 atm and a temperature of 100 to 600° C. to obtain a purified product, thereby removing amorphous carbon and producing the carbon nanotube product.