A general theory is developed for the reflection, transmission, and radiation of electromagnetic fields at the junction of two open waveguides. It employs the integral equation formulation for the field in the junction plane, in which point-source solutions in the two joining waveguides occur as kernel functions. The integral equations are solved by using the axial spectral representations of the kernel functions, together with the Fourier integral expansion with respect to the transverse coordinates. Numerical results are presented for a terminating slab waveguide, operating in the TE0 surface wave mode and radiating into free space.