Control is the process of influencing a dynamical system (by designing input functions or by choosing parameters) to make it behave in a desired way. This requires a deep understanding of the structural properties of the underlying system. In this talk, I will show how algebraic methods and symbolic computation can be used to gain insights into the system structure. In particular, I will discuss the problem of controlled and conditioned invariance for polynomially nonlinear control systems, as well as the rich theory of linear systems, where tools from homological algebra and module theory can be applied to characterize and classify system properties such as autonomy and controllability.