In this paper the irreversible behaviour of solids and structures in terms of rate-independent elastoplastic constitutive models in the presence of uncertainty in both material description and loading is studied. The mathematical background in convex analysis of deterministic elastoplasticity is extended to the stochastic domain, and numerical algorithms are formulated and explored. Computationally these problems—in analogy to the deterministic closest point return map—lead to the stochastic minimisation of a convex energy functional on tensor product spaces. The material parameters describing the problem are modelled as tensor-valued random fields whose numerical representation can be achieved in different ways, such as sampling, or with functional approximations using the Karhunen–Loève and polynomial chaos expansions. Various numerical solution strategies such as direct integration, stochastic Galerkin and collocation approaches are formulated, discussed, and compared on computational examples.