Distributionally robust stochastic optimization (DRSO) is a framework for decision-making problems under certainty, which finds solutions that perform well for a chosen set of probability distributions. Many different approaches for specifying a set of distributions have been proposed. The choice matters, because it affects the results, and the relative performance of different choices depend on the characteristics of the problems. In this paper, we consider problems in which different random variables exhibit some form of dependence, but the exact values of the parameters that represent the dependence are not known. We consider various sets of distributions that incorporate the dependence structure, and we study the corresponding DRSO problems. In the first part of the paper, we consider problems with linear dependence between random variables. We consider sets of distributions that are within a specified Wasserstein distance of a nominal distribution, and that satisfy a second-order moment constraint. We obtain a tractable dual reformulation of the corresponding DRSO problem. This approach is compared with the traditional moment-based DRSO, which considers all distributions whose first- and second-order moments satisfy certain constraints, and with the Wasserstein-based DRSO, which considers all distributions that are within a specified Wasserstein distance of a nominal distribution (with no moment constraints). Numerical experiments suggest that our new formulation has superior out-of-sample performance. In the second part of the paper, we consider problems with various types of rank dependence between random variables, including rank dependence measured by Spearman's footrule distance between empirical rankings, comonotonic distributions, box uncertainty for individual observations, and Wasserstein distance between copulas associated with continuous distributions. We also obtain a dual reformulation of the DRSO problem. A desirable byproduct of the formulation is that it also avoids an issue associated with the one-sided moment constraints in moment-based DRSO problems.