Mathematics plays a fundamental role in many of our activities which is often unappreciated. The course will look at a selection of use cases where mathematics is employed in design and optimisation. We will look at the details of the mathematics: what it does, what it's limitations are, why we rely on it and why an industry might fund mathematical research. The lectures will aim for a degree of rigour and avoid hand-waving. The examples will cover a wide range of topics from: Fluid Flow, Cryptography, Aircraft Icing, Probability and Uncertainty. The target audience is those interested in what mathematics can really do and how it helps us all.