In this report shows that a twist map of an annulus with a periodic point of rotation number p/q must have a Birkhoff periodic point of rotation number p/q. Topological techniques are used so no assumption of area-preservation or circle intersection property is needed. If the map is area preserving then this theorem and the fixed point theorem of Birkhoff imply a recent theorem of Mather. It is also shown that periodic orbits of (significantly) smallest period for a twist map must be Birkhoff. (Author)