From computation in neural networks to allostery in proteins, numerous natural and artificial systems are comprised of many interacting parts that give rise to advanced functions. To study such complex systems, a diverse array of interdisciplinary tools have been developed that relate the interactions of existing systems to their functions. However, engineering the interactions to perform designed functions in novel systems remains a significant challenge due to the nonlinearities in the interactions and the vast dimensionality of the design space. Here we develop design principles for complex dynamical and mechanical systems at the lowest level of their microstate interactions. In dynamical neural systems, we use methods from control theory and dynamical systems theory to mathematically map precise patterns of neural connectivity to the control of neural states in human and non-human brains (Chapter 2) and to the learning of computations on internal representations in artificial recurrent neural networks (Chapter 4). In mechanical systems, we use methods from algebraic geometry and dynamical systems to mathematically map precise patterns of mechanical constraints to design shape changes as a minimal model of protein allostery and cooperativity (Chapter 6) and to engineer mechanical metamaterials that possess arbitrarily complex shape changes (Chapter 8). These intuitive maps allow us to navigate previously unexplored design spaces in nonlinear and high-dimensional regimes, enabling us to reverse engineer form from function in novel complex systems that have yet to exist.