The author studies \(\beta\)-expansions in algebraic function fields over finite fields. More precisely, let \({\mathbb F}\) be a finite field, \({\mathbb F}[x]\) be its ring of polynomials, and \({\mathbb F}((x^{-1}))\) be its field of formal Laurent series. Given \(z\) and \(\beta\) in \({\mathbb F}((x^{-1}))\), say that \[ z=\sum_{i=1}^{\infty}{{d_i}\over {\beta^i}} \] is a representation of \(z\) in base \(\beta\) if \(d_i\in {\mathbb F}[x]\) for \(i=1,2,\ldots\). In this paper, the author defines Pisot and Salem elements of \({\mathbb F}((x^{-1}))\) in a manner reminiscent to the way they are defined in the field of real numbers. He then studies the set of \(z\in{\mathbb F}((x^{-1}))\) having periodic base \(\beta\)-expansions (denoted by \(\text{Per}(\beta)\)) and finite \(\beta\)-expansions (denoted by \(\text{Fin}(\beta)\)), respectively. Here are some of the results in the paper. Theorem 4.1: If \(\beta\) is Pisot or Salem, then \(\text{Per}(\beta)={\mathbb F}(x,\beta)\). This theorem can be seen as a function field analogue of a result proved independently by Bertrand and Schmidt in the late 70's to the effect that real numbers with periodic \(\beta\) expansions are precisely the positive real numbers in \({\mathbb Q}(\beta)\) once \(\beta\) is a Pisot number. Theorem 4.4: If \({\mathbb F}[x]\in \text{Per}(\beta)\), then \(\beta\) is Pisot or Salem. Theorem 5.2: If \(\beta\) is Pisot, then \(\text{Fin}(\beta)={\mathbb F}[x,\beta^{-1}]\). Theorem 5.4: If \({\mathbb F}[x,\beta^{-1}]\subset \text{Fin}(\beta)\), then \(\beta\) is Pisot.