Simulation models often lack tractable likelihood functions, making likelihood-free inference methods indispensable. Approximate Bayesian computation generates likelihood-free posterior samples by comparing simulated and observed data through some distance measure, but existing approaches are often poorly suited to time series simulators, for example due to an independent and identically distributed data assumption. In this paper, we propose to use path signatures in approximate Bayesian computation to handle the sequential nature of time series. We provide theoretical guarantees on the resultant posteriors and demonstrate competitive Bayesian parameter inference for simulators generating univariate, multivariate, irregularly spaced, and even non-Euclidean sequences.