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This paper is a review of applications of delay differential equations to different areas of engineering science. Starting with a general overview of delay models, we present some recent results on the use of retarded, advanced and neutral delay differential equations. An emerging area for modeling with the help of delay equations is real-time dynamic substructuring, or hybrid testing. We introduce the main idea of this technique together with the latest advances. Special emphasis is given to the development of the theory and applications of partial delay differential equations. The review concludes with a summary of some open problems and questions concerning the analysis of spatially extended delayed systems.