Electromagnetic transmission through a filled slit in a conducting plane of finite thickness is investigated when the slit cross section is an arbitrary shape. A non-modal formulation is developed whereby the original problem is broken up, via the equivalence principle, into three isolated situations. An integral equation is written for each of these situations involving equivalent electric and magnetic currents as unknowns on the closed contour defining the slit cross section. These three integral equations are solved simultaneously and it is shown that they have a unique solution at all frequencies. A modal solution is also developed for the case when the slit cross section can be viewed as a sequence of two-dimensional rectangular cavities, each coupled to the other by an aperture. Results obtained from these two solutions are compared. Some additional slits, which cannot be solved by a modal solution, are investigated by the non-modal solution. Two approximate solutions are also investigated when the slit cross section is rectangular. For small slit widths w, such that kow 1, the transmitted fields can be written in terms of a slit impedance for the TE case, and a slit polarizability for the TM case. An equivalent circuit is developed for the TE case and comparison with the modal solution is made for the slit impedance and transmission coefficient. The equivalent circuit involving the aperture impedance of a flange is shown to accurately determine the transmission characteristics of the slit.