The goal of systems biology is to provide a predictive and quantitative understanding of complex biological systems and their experimental data using mathematical and computational models. Mathematical modelling and computational simulation has been a crucial tool for basic research in cardiac electrophysiology for decades, and has provided remarkable insights into many physiological mechanisms. More recently these quantitative cardiac models have begun to transition into safety-critical clinical and pharmaceutical development applications. Using cardiac mathematical models in such applications requires high levels of credibility as well as an accurate quantification of the uncertainty in the model predictions. In this thesis, we develop a suite of Bayesian inference and uncertainty quantification techniques and software tools to study cardiac cellular electrophysiology. We then design and perform high-throughput experiments to construct and validate cell-specific mathematical models for human Ether-à-go-go-Related Gene (hERG) channel dynamics, and to study its temperature dependence, which is important for cardiac safety pharmacology and cardiac action potential models. Using the Bayesian statistical framework together with the high-throughput measurements allows us to quantify the uncertainty and variability of the model parameters. We also model and analyse how experimental artefacts contribute to the observed variability of these recordings, allowing us to separate the effects of these artefacts from physiological behaviour. A similar approach for uncertainty characterisation and experimental design will have to be adapted for other ion currents to create better predictive models, as accurate ion channel models are essential for constructing trustworthy cardiac action potential models.