A new approach to the problem of identifying design feasibility and optimality under uncertainty is introduced. Based on the Bayesian concepts of predictive probability and expected utility, the method can quantify the feasibility of a process design and identify the optimal operation conditions when there are uncertainties in the process parameters. The use of Bayesian statistics enables the treatment of a very wide class of parameter uncertainties, including simple bounds, analytic probability density functions, correlation structures and empirical distributions. Numerical solution of the feasibility and optimization problems is accomplished using Markov Chain Monte Carlo (MCMC) numerical techniques. Examples used to illustrate the new approach are drawn from: chemical kinetics, process operability and the design of unit operations where the information on the model parameters is obtained from statistical analysis of experimental data.