Solitary water waves are traveling waves that are localized in space and maintain their shape as they propagate. They are among the most fundamental structures in fluid mechanics and have inspired a large body of work ranging from formal asymptotics to rigorous analysis. In this talk, I will discuss several methods for constructing solitary water waves. I will then sketch a recent work in which we constructed solitary-wave solutions to the two-dimensional finite-depth water-wave equations with capillarity and constant vorticity. Part of this talk is based in part on joint work with Ting-Yang Hsiao, Giang To, and Ye Zhang.