This work focuses on mathematical methods and modern symbolic computational tools required to solve fundamental and advanced problems in anisotropic elasticity. Specific applications are presented to the class of problems that are encountered in the theory. Key features: particular emphasis is placed on the selection of analytic methodology for a specific problem and the potential of symbolic computational techniques to support and develop the analytic approach to problem-solving · the physical interpretation of exact and approximate mathematical solutions is thoroughly examined and provides new insights into the involved phenomena · state-of-the-art solutions are provided for a wide range of composite material configurations developed by the authors, including nonlinear problems and advanced analysis of laminated and thin-walled structures · ample graphic examples, including animations, further facilitate an understanding of the main steps in the solution process.