From a sequential testing perspective, the Wiener hypothesis testing problem aims at finding the correct hypothesis on the unknown drift of an observed Wiener process. Among given hypotheses, one is correct, and using the observations from the process, we would like to identify the correct one as quickly as possible. In this talk, we will discuss some different formulations of this problem, and we will present some of our related contributions in the literature. These contributions are based on joint work Savaş Dayanık from Bilkent University.