Each chapter of this thesis is represented as a metaphorical meeting between mathematicians and their ideas: Stieltjes meet Gauss; Berg meet Urbanik; Dynkin meet Jacobi; Dynkin meet Villani. In the "Stieltjes meet Gauss'' part we provide some new criteria for the determinacy problem of the Stieltjes moment problem, starting with a Tauberian type criterion for indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for determinacy. We also provide a criterion for determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an Abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to Pedersen's refinement of Krein's celebrated theorem. The proofs utilize non-classical Tauberian results for moment sequences that are analogues of the ones developed by Feigin and Yaschin and Balkema et al. for the bi-lateral Laplace transform in the context of asymptotically parabolic functions, which generalize the classical Gaussian setting. We illustrate these results by studying the time-dependent moment problem for the law of a process whose logarithm is a Lévy process, which is a generalization of the log-normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally-negative Lévy processes having a Gaussian component, which may be of independent interest. We continue the study of this time-dependent moment problem in the "Berg meet Urbanik'' part where we focus on Berg-Urbanik semigroups, a class of multiplicative convolution semigroups on $\R_+$ that is in bijection with the set of Bernstein functions. Berg and Durán proved that the law of such semigroups is determinate ...