The usual textbook treatment of electromagnetic (em) wave propagation across the boundary separating two dielectrics is generalized to materials characterized by complex refractive indices. Such materials are being investigated intensively and possess interesting and potentially useful properties in the infrared (IR) and optical bands. The sum of the transmission and reflection coefficients at the boundary in these materials does not add to unity, as in the case of perfect dielectrics, and the reason for this unexpected behavior is examined. Since these materials are usually fabricated in film form, appropriate transmission and reflection coefficients of thin and thick films with arbitrary refractive indices are derived (taking account of multiple reflections) to point out the differences introduced by the complex nature of the refractive index which have not been included in previous results or which have generally not been emphasized. Results are also presented for films deposited on thick substrates of arbitrary refractive indices since this is the common mode of fabrication. Graphs are presented to illustrate the transmission behavior of a thin absorbing film with and without a substrate.