The idea that democracy is under threat, after being largely dormant for at least 40 years, is looming increasingly large in public discourse. Complex systems theory offers a range of powerful new tools to analyse the stability of social institutions in general, and democracy in particular. What makes a democracy stable? And which processes potentially lead to instability of a democratic system? This paper offers a complex systems perspective on this question, informed by areas of the mathematical, natural, and social sciences. We explain the meaning of the term 'stability' in different disciplines and discuss how laws, rules, and regulations, but also norms, conventions, and expectations are decisive for the stability of a social institution such as democracy.