An arbitrarily shaped aperture in the wall of a uniform waveguide of unspecified cross-section has been analysed in terms of its scattering and radiating properties, using variational techniques. The method is of general application, the principal requirement being that the Green's functions for the waveguide and the external medium or structure should be known. Attention is focused particularly on the problem of radiation into free space from a nonresonant narrow rectangular slot arbitrarily located in the broadwall of plane rectangular waveguide. Analytical expressions are presented for the transmission and reflection coefficients and the power radiated, and, where possible, the results are compared with those of other theories. An expression for the electric field in the slot has been derived, and this has been used to formulate rigorous conditions under which circular polarisation is radiated by crossed slots. The generally expressed conditions for circular polarisation are shown to be approximate only.