The complementary nature of physically-based and data-driven models in their demand for physical insight and historical data, leads to the notion that the predictions of a physically-based model can be improved and the associated uncertainty can be systematically reduced through the conjunctive use of a data-driven model of the residuals.  The objective of this thesis is to minimise the inevitable mismatch between physically-based models and the actual processes as described by the mismatch between predictions and observations.   Principles based on information theory are used to detect the presence and  nature of residual information in model errors that might help to develop a data-driven model of the residuals by treating the gap between the process and its (physically-based) model as a separate process.  The complementary modelling approach is applied to various hydrodynamic and hydrological models to forecast the expected errors and accuracy, using neural network and fuzzy rule-based models.  Complementary modelling offers the opportunity of incorporating processes and data that are not considered by the model, without affecting the routine operation of physically-based models.  The possibility that information may be obtained which will help to improve the physically-based model is also demonstrated.