High-fidelity simulation of observational data given parameters is standard across scientific disciplines, but scientific discovery usually requires statistical inference to solve the inverse problem. Despite accurately modelling the data, complex simulators are often poorly suited to this task because the reverse mapping of observational data to parameters is usually unspecified and intractable. Density and density-ratio estimators can learn a probabilistic solution to the inverse problem using samples of parameters from a distribution of prior beliefs together with simulated data, so-called Simulation-based Inference. The aim of the PhD project is to develop trustworthy and accurate inference methods featuring a learned surrogate model of the solution to the inverse problem with complex simulators. Applications explored in the PhD leverage sophisticated simulators for gravitational waves, exoplanet transits, and the cosmic microwave background.