The course title, approximation theory, covers a great deal of mathematical territory. In the present context, the focus is primarily on the approximation of real-valued continuous functions by some simpler class of functions, such as algebraic or trigonometric polynomials. Such issues have attracted the attention of thousands of mathematicians for at least two centuries now. We will have occasion to discuss both venerable and contemporary results, whose origins range anywhere from the dawn of time to the day before yesterday.