In this commentary we are concerned with mathematical models of phosphorus transfers from agricultural land, particularly intensive grassland, to receiving waters (Haygarth et al., 2006). We believe that the complexity of this problem requires an initially wide discussion of possible modelling concepts and the associated benefits or problems with these approaches. Hence we are reluctant to break the phosphorus-modelling problem down to smaller scales or detailed levels of complexity, e.g. sediments/colloids, which is the main concern of previous papers in this series of commentaries. However, many aspects of the following discussion could be applied equally to sub-elements of the phosphorus-modelling problem as well as wider environmental modelling applications.Two objectives justify the building of mathematical phosphorus transfer models: (1) they formalize our knowledge about the system under study and in this way they act as vehicles for testing our understanding of phosphorus transfers; and (2) they may allow quantitative predictions under changing and complex environmental conditions. In the light of international legislation such as the EU Water Framework Directive (2000/60/EC), it is increasingly important to predict the impacts of land management practises and measures for mitigating diffuse pollution. This, however, requires a confidence in the process formulation (be it empirically or physically-based) and a predictive capability of models, which may not be justified in many cases, especially where there are limited observational data available to develop, test and drive model assumptions.In the previous papers of this series of commentaries, we learned that our perceptual understanding of phosphorus, sediment and colloid transfers from intensive grasslands is far from perfect (Bilotta et al., 2007; Gimbert et al., 2007; Granger et al., 2007; Haygarth et al., 2006). However, this perceptual understanding is (and will always be) better than our translation of these perceptions into conceptual and, finally, procedural models. 1 Two main approaches to modelling the phosphorus transfer problem can be identified from the literature: (1) empirical phosphorus export models to calculate annual or seasonal loads (this includes empirical risk assessment techniques); and (2) process-orientated phosphorus models to simulate storm dynamics. This includes conceptual and physically-based models 2 (Table I). In the following sections, we highlight a few critical points regarding these approaches and, finally, 1 Here we refer to the three classic steps of model development: the 'perceptual model' is our qualitative understanding of system behaviour, the 'conceptual model' comprises the mathematical equations to describe this behaviour, and the 'procedural model' is the computer code to run the actual simulations. Note the term 'conceptual model' is also used with a different notion later in this paper, referring to a mathematical model class of intermediate complexity. 2 The classification of models by the degree of c...