This chapter focuses on asymmetric random walks on the integers. It begins with questions of recurrence and transience, as well as the distribution of first-passage times, where generating functions prove to be a powerful tool. Next, it examines first return times and the geometrically distributed number of zeros in the case of an asymmetric random walk. Subsequent topics include random walks with absorbing boundaries, with an application to the gambler’s ruin problem, as well as the longest upward and downward runs. The chapter concludes with the Galton-Watson process, which serves as the simplest model for stochastic population dynamics. An important question in this context is the probability of such a process going extinct.