This is a three-year REU Program, with the purpose of motivating and training undergraduate students with majors in the applied sciences to use mathematical ideas and techniques to study and solve real-life physical and biological problems. The PIs and senior personnel will work with these students on a range of applied mathematics problems, using a combination of modeling, analysis --- especially of (nonlinear) partial differential equations (PDE) and calculus of variations --- numerical simulations, and hands-on experiments. We have brought together a balanced and diverse group for this proposal, including two researchers with expertise in PDEs, two in numerical analysis, and two experimentalists. Moreover all members of our group are experienced in the two unifying themes of this proposal: mathematical modeling and research training of undergraduates. A key aspect of this proposal is the fundamental role played by the concurrent and interdependent training of students in modeling, analysis, simulation, and experiments. Such training will provide a much needed opportunity to prepare the new workforce in science and engineering.