This is a self-contained introduction to algebraic functions and projective curves. The author covers a wide range of material by dispensing with the machinery of algebraic geometry and proceeding directly via valuation theory to the main results on function fields. The author also develops the theory of singular curves by studying maps to projective space, including topics such as Weierstrass points in characteristic p, and the Gorenstein relations for singularities of plane curves. The book is based on a graduate course taught by the author.