The present invention relates to an apparatus and method for measuring the critical current of a superconducting tape. A continuous critical current measurement apparatus for measuring critical current of a superconducting tape while feeding a superconducting tape in a liquid nitrogen container includes wheel-type current terminals and wheel-type voltage terminals. The superconducting tape is continuously supplied and fed by a reel-to-reel device, and the critical current of the superconducting tape is measured in real time using the wheel-type current terminals and the wheel-type voltage terminals while the superconducting tape is fed at constant linear velocity in contact with the wheel-type current terminals and the wheel-type voltage terminals. Accordingly, current is applied using wheel-type current terminals and voltage is measured using wheel-type voltage terminals while a superconducting tape is continuously supplied by a reel-to-reel device, thus continuously measuring critical current without burning out the superconducting tape.