The linear-quadratic model is one of the key tools in radiation biology and physics. It provides a simple relationship between cell survival and delivered dose: S=exp(-αD-βD2), and has been used extensively to analyse and predict responses to ionising radiation both in vitro and in vivo. Despite its ubiquity, there remain questions about its interpretation and wider applicability—Is it a convenient empirical fit or representative of some deeper mechanistic behaviour? Does a model of single-cell survival in vitro really correspond to clinical tissue responses? Is it applicable at very high and very low doses? Here, we review these issues, discussing current usage of the LQ model, its historical context, what we now know about its mechanistic underpinnings, and the potential challenges and confounding factors that arise when trying to apply it across a range of systems.