Inverse problems, broadly defined as the task of estimating unknown input parameters of mathematical models from observed data, arise across a wide range of scientific and engineering disciplines. This talk presents deep generative approaches to solving such problems within a Bayesian inference framework, covering two complementary settings distinguished by whether the likelihood function is tractable. In the first half, we address the tractable likelihood setting, where Markov chain Monte Carlo (MCMC) has long served as the standard inference tool but suffers from slow mixing and high computational cost. We propose replacing MCMC with normalizing flow-based variational inference, which leverages GPU computing for substantially faster posterior approximation. We show, however, that naïve application of normalizing flows is insufficient: accurate posterior representation requires careful architectural choices—including mixture-based distributions to handle multimodality and tail-adaptive transformations to capture heavy-tailed behavior—as well as principled training strategies such as weight-adjusted fine-tuning to mitigate the mode-seeking bias of reverse KL divergence. In the second half, we turn to the intractable likelihood setting, where complex, high-dimensional, or semi-continuous data structures (such as spatial fields with excessive zeros) preclude explicit likelihood evaluation. Here, we employ denoising diffusion probabilistic models (DDPM) as emulators of the computer model output, and combine them with approximate Bayesian computation (ABC) in which a Siamese network extracts discriminative features to compute data-adaptive acceptance probabilities. Together, these methods extend the reach of principled Bayesian calibration to a broader class of scientifically important models.