The overall theme of this report is that of algebraic function fields. Specifically, we examine several bounds on the number of rational places of function fields over finite fields. The necessary tools for discussing such bounds are introduced in the first chapter. This includes places, valuations, divisors and Riemann-Roch spaces. We also define Weierstraß semigroups and relate them to function fields. We then undertake the main task of the report by presenting and assessing five different bounds. Our point of departure is the Hasse-Weil bound, which is swiftly improved upon by Serre. We then assume further knowledge of our function field in order to examine the Lewittes and Geil-Matsumoto bound, the latter of which is further generalised by Beelen and Ruano. Lastly, we apply the bounds on several families of function fields, for which we mention some of their known properties. We compare the results of the bounds for each function field by examples.