Two-periodic random walks are models for the one-dimensional motion of particles in which the jump probabilities depend on the parity of the currently occupied state. Such processes have interesting applications, for instance in chemical physics where they arise as embedded random walk of a special queueing problem. In this paper we discuss in some detail first passage time problems of two-periodic walks, the distribution of their maximum and the transition functions when the motion of the particle is restricted by one or two absorbing boundaries. As particular applications we show how our results can be used to derive the distribution of the busy period of a chemical queue and give an analysis of a somewhat weird coin tossing game. / Series: Research Report Series / Department of Statistics and Mathematics