Brittle materials are characterized by a wide scatter in their strength data and by size effects. In this work, size effects in brittle composite structures such as beams and plates with and without stress raisers are investigated experimentally and analytically. For structures with geometrical irregularities such as notches, holes or cut-outs, no analytical solutions exist and approximate numerical schemes have to be employed. For this purpose, the finite element method is extensively used. For stress analysis of isotropic beams under four point bending and composite plates under in-plane loading, a finite element analysis program F2DELAST based on two-dimensional elasticity theory was developed and verified with existing literature and also by a commercial finite element code ABAQUS. For composite beams under four-point bending, ABAQUS is used. For stress and failure analysis of laminated composite plates under transverse loading, a nonlinear finite element analysis program ENCOMPLIT which accounts for the geometrical nonlinearity in von Karman sense and transverse shear deformation was developed. Both the finite element programs written by the author and the ABAQUS were used for stress analysis as convenience dictates. For post-processing purposes, PATRAN and I-DEAS are used. Linear and nonlinear stress analyses were performed for the case of composite plates under transverse loading. A methodology for reliability analysis is developed for composite beams and plates with and without stress concentration under various loading conditions and various finite element reliability analysis programs were written. The materials used for the analysis are mainly carbon-carbon composites, however, where validation of finite element codes or comparison of the proposed failure criteria with other criteria are needed, graphite-epoxy composites, which are extensively used in the literature, are also used. The model is based on Weibull statistics. The Weibull model was extended to account for the three-dimensional stress field including interlaminar stresses. The effect of interlaminar stresses on the total reliability is investigated. Various failure theories for composite materials are reviewed. New failure criteria for bi-directional orthotropic lamina are developed and proposed. Failure analysis in the non-probabilistic sense (i.e. First-ply-failure analysis) is performed on carbon-carbon composite beams under 4-point bending and carbon-carbon composite plates under transverse loading using the existing failure criteria and the failure criteria proposed in this work. Comparisons are made with the existing failure criteria and also with experiments. An attempt was made to relate the (probabilistic) reliability analysis with (non-probabilistic) first-ply-failure analysis in terms of failure loads for composite beams under 4-point bending and square and rectangular composite plates under transverse loading.Ph. D.