An algebraic function field is an algebraic extension of finite degree over the rational function field. This field was first introduced by Dedekind, Kronecker, and Weber in the nineteenth century and was further developed by Artin, Hasse, Schmidt, and Weil in the first half of the twentieth century. In this talk, we discuss various function field extensions over a finite field and examine important invariants such as the number of rational places and the class numbers.