Abstract: Tackling the immediate challenges that arise from growing model complexities (spatiotemporal measurements) and data-intensive studies (large-scale and high-dimensional measurements collected as time-series), state-of-the-art methods can easily exceed their limits of applicability. Conventional methods based on Gaussian processes (GP) often fall short in providing satisfactory solutions since they tend to offer over-smooth priors. Recently, the Besov process (BP), defined by wavelet expansions with random coefficients, has emerged as a more suitable prior for Bayesian inverse problems of this nature. While BP excels in handling spatial inhomogeneity, it does not automatically incorporate temporal correlation inherited in the dynamically changing objects. In this talk we describe how to generalize BP to a novel spatiotemporal Besov process (STBP) by replacing the random coefficients in the series expansion with stochastic time functions as Q-exponential process which governs the temporal correlation structure. We thoroughly investigate the mathematical and statistical properties of STBP. A transport map representation of STBP is also proposed to facilitate the inference. Towards the end of the talk, we will mention two alternatives for Bayesian inference of dynamic inverse problems with heavy-tailed priors--one based on a goal-oriented approach to overcome high computational cost and the other approach based on sparse Bayesian learning that allows automatic parameter estimation. We demonstrate the utility of our approach on several image reconstruction problems. This is based on joint work with (Shiwei Lan, Shuyi Li, Weining Shen) and (Jonathan Lindbloom, Jan Glaubitz, Youssef Marzouk).