Distributionally robust optimization is a modeling paradigm for optimization under uncertainty where the distribution of the uncertain parameters is ambiguous, and where one seeks decisions that minimize the worst-case expected cost with respect to all distributions in a prescribed ambiguity set. In this paper, we develop a general theory of nonlinear distributionally robust optimization using the language of convex analysis. We study static minimization problems whose cost functions are convex in the decision variables and piecewise concave in the uncertain parameters, while the ambiguity set contains all distributions that satisfy several conditions involving convex moment functions. Leveraging a generalized `primal worst equals dual best' duality scheme for robust optimization, we derive from first principles a strong duality result that relates distributionally robust to classical robust optimization problems and that obviates the need to mobilize the machinery of abstract semi-infinite duality theory. We also derive finite convex reformulations for nonlinear distributionally robust optimization problems. We illustrate the modeling power of the proposed approach through convex reformulations for data-driven distributionally robust optimization problems whose ambiguity sets constitute type-$p$ Wasserstein balls for any $p \in [1,\infty]$, thus complementing the only known reformulations for $p \in \{1,\infty\}$.

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