Recovering piecewise-smooth functions with jump discontinuities from noisy, indirect, and incomplete data is a central task in computational inverse problems. Bayesian neural networks (BNNs) with heavy-tailed weight priors have emerged as effective edge-preserving priors for such problems. However, posterior inference for these models remains computationally challenging, and existing approaches settle for point estimates obtained by directly optimizing heavy-tailed log-densities. In our recent preprint, we propose GSBL-BNNs, which model the neural network parameters hierarchically using sparsity-promoting priors from generalized sparse Bayesian learning (GSBL). The hierarchical formulation recovers several popular marginal priors, including Laplace and Student's t (including Cauchy). A key modeling ingredient, novel among edge-preserving BNNs and native to the hierarchical formulation, couples each neuron's bias to its weights through a shared hyperparameter, allowing jump discontinuities anywhere in the domain while preserving the conditionally Gaussian structure underlying efficient inference. We developed an efficient block-coordinate descent (BCD) algorithm for maximum a posteriori (MAP) estimation. We perform approximate Bayesian inference and uncertainty quantification (UQ) using deep ensembles and Markov chain Monte-Carlo (MCMC) methods. Computational experiments---including signal denoising, deblurring, and inverting a two-dimensional hyperbolic conservation law--- show that GSBL-BNNs deliver accurate, edge-preserving reconstructions with built-in UQ. Compared with Gaussian- and Cauchy-BNNs, they yield more accurate predictive means and avoid overconfident uncertainty estimates. My main supervisor is Jan Glaubitz from the Division of Applied Mathematics at Linköping University. You can find his contact details here: https://www.janglaubitz.com/