Rational decision making (for flood warning, navigation, or reservoir systems) requires that the total uncertainty about a hydrological predictand (such as river stage, discharge, or runoff volume) be quantified in terms of a probability distribution, conditional on all available information and knowledge. Hydrological knowledge is typically embodied in a deterministic catchment model (lumped, semi-distributed, or distributed). Fundamentals are presented of a Bayesian Forecasting System (BFS) for producing a probabilistic forecast of a hydrological predictand via any deterministic catchment model. The BFS decomposes the total uncertainty into precipitation uncertainty and hydrologic uncertainty, which are modeled independently and then are integrated into a predictive (Bayes) distribution. The BFS is compared with Monte Carlo simulation and "ensemble forecasting" technique, none of which can alone produce a probabilistic forecast that meets requirements of rational decision making.