Discrete dynamical systems theory is concerned with the iteration (or repeated application) of functions from a space to itself. The theory has many different flavours, depending on the space concerned and on the properties which we assume the function to have (it might be complex analytic, differentiable, or only continuous). In this module we will be primarily concerned with the iteration of continuous functions defined on the interval or on the two-dimensional disk. Students who enjoyed MATH345 (The Magic of Complex Numbers – Complex Dynamics) are likely to appreciate this module, although MATH345 is not a prerequisite and there will be little overlap between the modules except for basic definitions and examples. The section on one-dimensional dynamics will develop some of the themes introduced in MATH345, including a proof of Sharkovsky's famous theorem on the possible sets of periods of continuous self-maps of the real line. The material on surface dynamics will culminate with a study of the dynamical implications of Thurston's magnificent classification theorem for isotopy classes of surface homeomorphisms.