Applied mathematics plays a major role in the sciences and in engineering, where it enhances our ability to use models and handle data. While pure mathematics mostly focuses on abstract structures, applied mathematics provides a tool for understanding and tackling a multitude of practical problems, such as medical imaging and neuroscience. Here, Alain Goriely explains the nature of applied mathematics, and discusses its early achievements in physics adn engineering, and its development as a separate field after World War II. Using historical examples, and current applications and challenges, Goriely illustrates the role that mathematics plays in the modern sciences today, and its far-reaching potential.