I gave a tutorial on "Introduction to Bayesian Neural Networks" at the 2022 Nordic Probabilistic AI School (ProbAI 2022), where I mainly talked about how to apply some popular VI techniques for Bayesian Neural Networks, with some hands-on demos.

Tutorial notebooks: regression (answers) classification (answers)

For Laplace Approximation, also see Jose Miguel Hernandez-Lobato's tutorial on modern Laplace methods applied to Bayesian Neural Networks.