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Much has been said about the fusion of bio-inspired optimization algorithms and Deep Learning models for several purposes: from the discovery of network topologies and hyperparametric configurations with improved performance for a given task, to the optimization of the model's parameters as a replacement for gradient-based solvers. Indeed, the literature is rich in proposals showcasing the application of assorted nature-inspired approaches for these tasks. In this work we comprehensively review and critically examine contributions made so far based on three axes, each addressing a fundamental question in this research avenue: (a) optimization and taxonomy (Why?), including a historical perspective, definitions of optimization problems in Deep Learning, and a taxonomy associated with an in-depth analysis of the literature, (b) critical methodological analysis (How?), which together with two case studies, allows us to address learned lessons and recommendations for good practices following the analysis of the literature, and (c) challenges and new directions of research (What can be done, and what for?). In summary, three axes – optimization and taxonomy, critical analysis, and challenges – which outline a complete vision of a merger of two technologies drawing up an exciting future for this area of fusion research. • We thoroughly examine the fusion between Deep Learning and bioinspired optimization. • Definitions and a taxonomy of Deep Learning optimization problems are provided. • We perform a critical methodological analysis of contributions made so far. • Learned lessons and recommendations are drawn from our analysis and two study cases. • Challenges and research directions are given in this fusion of technologies. [ABSTRACT FROM AUTHOR]