Periodic structures, such as frequency-selective surfaces (FSSs) and photonic band-gap (PBG) materials, exhibit total reflection in specific frequency bands while total transmission in other bands. They find numerous applications in a large field of the electromagnetic (EM) spectrum. For example, in the microwave region, they are used to increase the efficiency of reflector antennas. In the far-infrared region they are used in designing polarizers, beam splitters, mirrors for improving the pumping efficiency in molecular lasers, as components of infrared sensors, etc. To set a solid basis for the analysis of periodic structures, we have first studied the most commonly used technique, the integral equation (IE) solved by the method of moments (MoM). IE-MoM is particularly well-suited for the analysis of printed planar structures. In any IE-MoM numerical implementation the efficient evaluation of the corresponding Green's functions (GFs) is of paramount importance. This is especially true for IE analysis of periodic structures whose GFs are slowly converging infinite sums. The systematic study of existing acceleration algorithms of general and specific types, used to accelerate the evaluation of periodic GFs, has been performed. We propose a new and efficient method for acceleration of multilayered periodic GFs that successfully combines the advantages of Shanks' and Ewald's transform. In structures operating at higher frequencies (thin films in millimeter and submillimeter wave bands or with self supporting metallic plates) the thickness of metallic screens must be taken into account. The existing full-wave approaches for simulating these structures double the number of unknowns as compared to that one of the zero-thickness case. Moreover, the thick aperture problem asks for the computation of cavity Green's functions, which is a difficult and time-consuming task for apertures of arbitrary cross-sections. This thesis addresses the problem of scattering by periodic apertures in conducting screens of finite thickness by introducing an approximate and computationally efficient formulation. This formulation consists in treating the thick aperture as an infinitely thin one and in using the correction term in integral equation kernel that accounts for the screen thickness. The number of unknowns remains the same as in the zero-thickness screens and evaluation of complicated cavity Green's functions is obviated, which yields computationally efficient routines.