The arithmetic dynamics of rational functions have been studied in many contexts. In this thesis, we concentrate on periodic points. For \phi(x)=x^2+c with c rational, we give a parametrization of all points of order 4 in quadratic fields. For a point of order two and a point of order three for a rational function defined over a number field with good reduction outside a set S, it is known that the bilinear form B([x_1, y_1], [x_2, y_2]) = x_1y_2 - x_2y_1 yields a unit in the ring of S-integers of a number field. We prove that this is essentially the only bilinear form with this property. Finally, we give restrictions on the orders of rational periodic points for rational functions with everywhere good reduction.