Abstract: Aircraft flight planning is impacted by weather uncertainties. Existing approaches to flight planning are either deterministic and load additional fuel to account for uncertainty, or probabilistic but have to plan in 4D space. If constraints are imposed on the flight plan these methods provide no guarantees that the constraints are actually satisfied. We investigate constrained flight planning under weather uncertainty on discrete airways graphs and model this problem as a Constrained Stochastic Shortest Path (C-SSP) problem. Transitions are generated on-the-fly by the underlying aircraft performance model. As this prevents us from using off-the-shelf C-SSP solvers, we generalise column-generation methods stemming from constrained deterministic path planning to the probabilistic case. This results in a novel method which is complete but computationally expensive. We therefore also discuss deterministic and heuristic approaches which average over weather uncertainty and handle constraints by scalarising a multi-objective cost function. We evaluate and compare these approaches on real flight routes subject to real weather forecast data and a realistic aircraft performance model.