A theoretical treatment of nuclear-magnetic double-irradiation experiments is given which is applicable to two groups of nuclei of arbitrary spin, either of different nuclear species or in a situation where the chemical shift is large in comparison with the spin-coupling constant. A convention is introduced whereby the group which experiences the ``strong'' rf field H2 is given the symbol X, while the group to be investigated is represented by A. Particular attention is given to groups of spin 1/2 nuclei in AnXm type molecules where m, n≤3, and their spin-decoupling behavior is presented in graphical form. Good correlations are observed with the experimental proton double-resonance spectra of acetaldehyde, diethyl succinate, and 1,1,2 trichloroethane. When a single X transition is irradiated with a very weak H2 a splitting of the A spin-multiplet lines into submultiplets is observed; at the same time an Overhauser-type redistribution of intensities may occur in certain molecules. Intermediate strengths of H2 centered on the X multiplet coalesce the A spectrum to what is essentially a single line for AXm molecules, but for AnXm molecules where n>1 there is a ``residual splitting'' which, although it decreases as H2 is made stronger, may never disappear completely. At high powers it is necessary to displace H2 slightly off resonance for group X towards the A resonance in order to obtain optimum decoupling, a correction which can be important in the accurate measurement of proton chemical shifts by spin-decoupling techniques.