This paper proposes distributed algorithms to solve robust convex optimization (RCO) when the constraints are affected by nonlinear uncertainty. We adopt a scenario approach by randomly sampling the uncertainty set. To facilitate the computational task, instead of using a single centralized processor to obtain a "global solution" of the scenario problem (SP), we resort to multiple interconnected processors that are distributed among different nodes of a network to simultaneously solve the SP. Then, we propose a primal-dual subgradient algorithm and a random projection algorithm to distributedly solve the SP over undirected and directed graphs, respectively. Both algorithms are given in an explicit recursive form with simple iterations, which are especially suited for processors with limited computational capability. We show that, if the underlying graph is strongly connected, each node asymptotically computes a common optimal solution to the SP with a convergence rate O(1/(Sigma(k)(t=1) zeta(t))), where {zeta(t)} is a sequence of appropriately decreasing stepsizes. That is, the RCO is effectively solved in a distributed way. The relations with the existing literature on robust convex programs are thoroughly discussed and an example of robust system identification is included to validate the effectiveness of our distributed algorithms.