This paper is about the existence of periodic points for homeomorphisms of the annulus and contains an affirmative answer to a question posed by \textit{G. R. Hall} [Contemp. Math. 81, 135--152 (1988; Zbl 0677.58026)]. Using Thurston-Nielsen theory for surface homeomorphisms, the following theorem is proved: If \(f\) is an orientation and boundary preserving homeomorphism of the annulus and if \(p/q\) is in the rotation set of \(f\) with \(p\) and \(q\) relatively prime, then \(f\) has a \((p,q)\) topologically monotone periodic orbit. If \(f\) has a \((p,q)\) orbit that is not topologically monotone, then the rotation set of \(f\) contains the Farey interval of \(p/q\).