Abstract: One of the major challenges in applied mathematics arises from the complexity of the problems that are being studied (e.g., try to predict the Earth's climate in the next 100 years). Not only that the equations cannot be solved by analytical methods, they can't even be solved numerically, since the number of required variables exceeds by much the computational resources. One is led to try and "simplify", or "reduce" the equations. In many cases, this is done in an ad-hoc, problem-specific manner, with little, or no mathematical justification; such approach is referred to as "modeling". In this talk I will describe the mathematical approach to model reduction, discuss various classes of problems in which rigorous model reduction is possible, and provide illustrative examples.