The use of computer simulations as models of real-world phenomena plays an increasingly important role in science and engineering. Such models allow us to build hypotheses about the processes underlying a phenomenon and to test them, e.g., by simulating synthetic data from the model and comparing it to observed data. A key challenge in this approach is to find those model configurations that reproduce the observed data. Bayesian statistical inference provides a principled way to address this challenge, allowing us to infer multiple suitable model configurations and quantify uncertainty. However, classical Bayesian inference methods typically require access to the model's likelihood function and thus cannot be applied to many commonly used scientific simulators. With the increase in available computational resources and the advent of neural network-based machine learning methods, an alternative approach has recently emerged: simulation-based inference (SBI). SBI enables Bayesian parameter inference but only requires access to simulations from the model. Several SBI methods have been developed and applied to individual inference problems in various fields, including computational neuroscience. Yet, many problems in these fields remain beyond the reach of current SBI methods. In addition, while there are many new SBI methods, there are no general guidelines for applying them to new inference problems, hindering their adoption by practitioners. In this thesis, I want to address these challenges by (a) advancing SBI methods for two particular problems in computational neuroscience and (b) improving the general applicability of SBI methods through accessible guidelines and software tools. In my first project, I focus on the use of SBI in cognitive neuroscience by developing an SBI method designed explicitly for computational models used in decision-making research. By building on recent advances in probabilistic machine learning, this new method is substantially more efficient than previous methods, allowing researchers to perform SBI on a broader range of decision-making models. In a second project, I turn to computational connectomics and show how SBI can help to discover connectivity rules underlying the complex connectivity patterns between neurons in the sensory cortex of the rat. As a third contribution, I help establish a software package to facilitate access to current SBI methods, and I present an overview of the workflow required to apply SBI to new inference problems as part of this thesis. Taken together, this thesis enriches the arsenal of SBI methods available for models of decision-making, demonstrates the potential of SBI for applications in computational connectomics, and bridges the gap between SBI method development and applicability, fostering scientific discovery in computational neuroscience and beyond.