This lecture will provide an overview of mathematical models, their types, and their construction. The general objective of mathematical modeling is to take a hypothesis, convert it a system of equations, and determine how well the equations describe reality. The specific objectives depend on the application, but could include predicting nutrient digestibility or intake. In this way, modeling is no different from any other scientific exercise—the first step is for the investigator to identify the hypothesis and objective. There are different types of models, and the investigator should choose a type suited to the specific objectives. In defining its type, a model can be categorized as static or dynamic, empirical or mechanistic, and deterministic or stochastic. Historically, nutrient requirement models have been static, empirical, and deterministic; they provided snapshots in time, did not describe mechanisms underlying responses, and did not consider inherent biological variance. These models were easy to derive, and have served the community well for more than a century. The Molly cow model is dynamic, mechanistic, and deterministic predicting responses through time based on underlying elements of digestion and metabolism without consideration of biological variation. After the investigator identifies the hypothesis, objective, and model type, the next step of constructing a model is to draw a block diagram. This diagram organizes the model conceptually. Rectangles in the diagram represent state variables, and arrows connecting the rectangles show the relationship of the variables. In a model of carbohydrate digestion in the rumen, for example, rectangles would represent pools of fiber, starch, and sugars, and an arrow connecting fiber and sugars would represent hydrolysis of fiber. This approach of representing pools or compartments within a system is referred to as compartmental modeling. In the remaining steps of constructing a model, the investigator translates the block diagram into system of equations, defines values of equation parameters, and solves the model so it can generate predictions. If evaluation of the model shows predictions are inadequate, earlier steps are repeated to refine the model.