This presentation will discuss various aspects of the modeling of physical, biological and engineering systems and how various areas of applied mathematics can contribute to modeling. Emphasis will be placed on the applied mathematics tools which one could use and the results which one could possibly expect to obtain with them. Topics to be covered include integrable evolution equations, variational approximations, numerics, symbolic computations, and computer simulations, with examples from physics, nonlinear optics and crowd dynamics, as well as general comments on the philosophy of mathematical applications.