Over the past 100years, deterministic rate equations have been successfully used to infer enzyme-catalysed reaction mechanisms and to estimate rate constants from reaction kinetics experiments conducted invitro. In recent years, sophisticated experimental techniques have been developed that begin to allow the measurement of enzyme-catalysed and other biopolymer-mediated reactions inside single cells at the single-molecule level. Time-course data obtained using these methods are considerably noisy because molecule numbers within cells are typically quite small. As a consequence, the interpretation and analysis of single-cell data requires stochastic methods, rather than deterministic rate equations. Here, we concisely review both experimental and theoretical techniques that enable single-molecule analysis, with particular emphasis on the major developments in the field of theoretical stochastic enzyme kinetics, from its inception in the mid-20th century to its modern-day status. We discuss the differences between stochastic and deterministic rate equation models, how these depend on enzyme molecule numbers and substrate inflow into the reaction compartment, and how estimation of rate constants from single-cell data is possible using recently developed stochastic approaches.