A general, linear theory of coupling electromagnetic fields from free-space waveguide arrays to an inhomogeneous plasma in a magnetic field is presented. In contrast to previous analyses, which assumed parallel-plate waveguides, waveguides having closed cross-sections of arbitrary shape are considered; full account is taken of all the waveguide modes. Far away from the coupling region, the plasma is assumed to be absorptive and the waveguides to propagate only their dominant modes. The formulation is geared to obtain the reflection coefficients in the waveguides and the excitation of waves in the plasma. The results are applicable to RF heating of plasmas whose size is large compared to the waveguide array, as would be the case in a fusion reactor, and valid for the frequency regimes of either ion-cyclotron (harmonic), or lower-hybrid, or electron-cyclotron (harmonic) heating.