Multidisciplinary design optimization (MDO) is a field of research that studies the application of numerical optimization techniques to the design of engineering systems involving multiple disciplines or components. Since the inception of MDO, various methods (architectures) have been developed and applied to solve MDO problems. This paper provides a survey of all the architectures that have been presented in the literature so far. All architectures are explained in detail using a unified description that includes optimization problem statements, diagrams, and detailed algorithms. The diagrams show both data and process flow through the multidisciplinary system and computational elements, which facilitates the understanding of the various architectures, and how they relate to each other.A classification of the MDO architectures based on their problem formulations and decomposition strategies is also provided, and the benefits and drawbacks of the architectures are discussed from both a theoretical and experimental perspective. For each architecture, several applications to the solution of engineering design problems are cited. The result is a comprehensive but straightforward introduction to MDO for non-specialists, and a reference detailing all current MDO architectures for specialists.