A non-gravity-type resistance exercise apparatus that includes a novel pulley-carrying exercise arm that can be incrementally rotated in a novel manner between high, low and numerous intermediate exercise positions to enable the proper performance of a number of different exercises. The apparatus includes a compact, self-contained, selectorized resistance module that embodies a plurality of discrete, elongated elastomeric cords that can be quickly and easily selectively coupled with the body-engaging means to provide precise resistance to the performance of several different kinds of exercises.