Rational decision making (for flood warning, navigation, or reservoir systems) requires that the total uncertainty about a hydrologic predictand (such as river stage, discharge, or runoff volume) be quantified in terms of a probability distribution, conditional on all available information and knowledge. Hydrologic knowledge is typically embodied in a deterministic catchment model. Fundamentals are presented of a Bayesian forecasting system (BFS) for producing a probabilistic forecast of a hydrologic predictand via any deterministic catchment model. The BFS decomposes the total uncertainty into input uncertainty and hydrologic uncertainty, which are quantified independently and then integrated into a predictive (Bayes) distribution. This distribution results from a revision of a prior (climatic) distribution, is well calibrated, and has a nonnegative ex ante economic value. 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, but each can serve as a component of the BFS.