Mathematics has an important place in society, but one that is not always obvious. ‘Mathematics, what is it good for? Quarternions, knots, and more DNA’ considers examples of mathematical theories—quarternions and knot theory—that have found unexpected applicability in sciences and engineering. Quaternions, introduced by William Rowan Hamilton in 1843 during his work on complex numbers, are now routinely used in computer graphics, robotics, missile and satellite guidance, as well as in orbital mechanics and control theory. Knot theory has found unexpected applications in the study of many physical and biological systems, including the study of DNA. Another example is the use of number theory in cryptography.