A major challenge in systems neuroscience is to understand how the dynamics of neural circuits give rise to behaviour. Analysis of complex dynamical systems is also at the heart of control engineering, where it is central to the design of robust control strategies. Although a rich engineering literature has grown over decades to facilitate the analysis of such systems, little of it has percolated into neuroscience so far. Here, we give a brief introduction to a number of core control-theoretic concepts that provide useful perspectives on neural circuit dynamics. We introduce important mathematical tools related to these concepts, and establish connections to neural circuit analysis, focusing on a number of themes that have arisen from the modern 'state-space' view on neural population dynamics.