Gamma-ray bursts (GRBs) are the most energetic explosions in the Universe. They are collimated, ultra-relativistic outflows which are powered by a central engine. When the ejected material starts to interact with the surrounding medium, shocks are formed where particles can be accelerated to emit synchrotron emission. This long-lived, broadband emission is called the afterglow of the GRB. The afterglow emission contains valuable information regarding the energetics of GRBs, dynamics of the ejected material, properties of the surrounding environment and how particles are accelerated in ultra-relativistic shocks. In this thesis, I investigate the physics of GRBs by modelling the observed afterglow emission using novel Bayesian inference techniques. We present a new approach to modelling GRB afterglow emission by making use of Gaussian processes (GPs) to take into account additional systematics in the data sets (Ch. 2) and we apply this new method to a sample of 26 GRBs to investigate how the parameters are distributed across the GRB population (Ch. 3). Furthermore, we perform a population synthesis study based on a flux-limited sample of GRBs with well-understood selection effects. Using Bayesian inference, we calibrate the parameter distributions of the synthetic population such that it recreates the observed properties of the GRB sample (Ch. 4). We also present radio observations of a short GRB provided by the Australia Telescope Compact Array rapid-response mode and we demonstrate how such observations allow us to constrain the physics of GRBs (Ch. 5). Finally, we present broadband afterglow modelling results for GRB 190114C, which is the first GRB detected at TeV energies (Ch. 6).