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Resonant tunneling through one-dimensional quantum-mechanical systems such as multibarrier structures and superlattices has been studied in order to solidify the foundation of exact and comprehensive treatment of it. For general systems with piecewise continuous potential and effective-mass profiles, it is proven that allowable energy levels for resonant tunneling comprise a band structure-i.e., resonance bands which are distinct, in general, from the so-called subbands. One can see a resonance is attained by adjusting external parameters to match with a real resonating pair of the objective section (internal structure). In a series connection of two or more subsections (or a heterostructure with two or more layers), resonant tunneling is shown to be caused, this time, by a matching of generally complex resonating pairs of each couple of succeeding subsections. Two particular kinds of structures, i.e., imaginal-iteration and simple-iteration structures, are investigated in detail with regard to the matching between subsections at each resonant energy level. Thus, the possibility of resonance can be seen not directly related with resonance bands of the subsections. The overall argument will provide insight into the mechanism of resonant tunneling