The need to model uncertainty in process design and operations has long been recognized. A frequently taken approach, the two-stage paradigm, involves partitioning the problem variables into two stages : those that have to be decided before and those that can be decided after the uncertain parameters reveal themselves. The resulting two-stage stochastic optimization models minimize the sum of the costs of the first stage and the expected cost of the second stage. A potential limitation of this approach is that it does not account for the variability of the second-stage costs and might lead to solutions where the actual second-stage costs are unacceptably high. In order to resolve this difficulty, we introduce a robustness measure that penalizes second-stage costs that are above the expected cost. Incorporating this measure into stochastic programming formulations does not introduce nonlinearlities, thus making possible the solution of large-scale problems through linear programming techniques. The proposed framework is applied to the planning of chemical process networks for which we propose a number of specialized models and solution schemes.