0 Preliminaries.- I Stochastic Two-Stage Problems.- 1 Convex Case.- 2 Nonconvex Case.- 3 Stability.- 4 Epi-Convergence.- 5 Saddle Property.- 6 Stochastic Independence.- 7 Special Convex Cases.- II Duality and Stability in Convex Optimization (Extended Results for the Saddle Case).- 8 Characterization and Properties of Saddle Functions.- 9 Primal and Dual Collections of Programs.- 10 Normal and Stable Programs.- 11 Relation to McLinden's Results.- 12 Application to Convex Programming.- III Barycentric Approximation.- 13 Inequalities and Extremal Probability Measures - Convex Case.- 14 Inequalities and Extremal Probability Measures - Saddle Case.- 15 Examples and Geometric Interpretation.- 16 Iterated Approximation and x-Simplicial Refinement.- 17 Application to Stochastic Two-Stage Programs.- 18 Convergence of Approximations.- 19 Refinement Strategy.- 20 Iterative Completion.- IV An Illustrative Survey of Existing Approaches in Stochastic Two-Stage Programming.- 21 Error Bounds for Stochastic Programs with Recourse (due to Kali & Stoyan).- 22 Approximation Schemes discussed by Birge & Wets.- 23 Sublinear Bounding Technique (due to Birge & Wets).- 24 Stochastic Quasigradient Techniques (due to Ermoliev).- 25 Semi-Stochastic Approximation (due to Marti).- 26 Benders' Decomposition with Importance Sampling (due to Dantzig & Glynn).- 27 Stochastic Decomposition (due to Higle & Sen).- 28 Mathematical Programming Techniques.- 29 Scenarios and Policy Aggregation (due to Rockafellar & Wets).- V BRAIN - BaRycentric Approximation for Integrands (Implementation Issues).- 30 Storing Distributions given through a Finite Set of Parameters.- 31 Evaluation of Initial Extremal Marginal Distributions.- 32 Evaluation of Initial Outer and Inner Approximation.- 33 Data forx-Simplicial Partition.- 34 Evaluation of Extremal Distributions - Iteration!.- 35 Evaluation of Outer and Inner Approximation - Iteration J.- 36 x-Simplicial Refinement.- VI Solving Stochastic Linear Two-Stage Problems (Numerical Results and Computational Experiences).- 37 Testproblems from Literature.- 38 Randomly Generated Testproblems.