This mini-course aims to give an introduction to the subject of relations between anti-self-dual equations and integrable systems.  It is designed to be an introductory course for first-year graduate students, starting with a summary of mathematical background required to define the anti-self-dual Yang-Mills equation in four dimensions.  Then the integrability of the equation is discussed, with ideas of the twistor correspondence presented.  The course is planned to end with a brief overview of the anti-self-dual condition for conformal structures and related integrable systems.