Open channel hydraulics, while rooted in civil and environmental engineering, ultimately centers on the study of first-order partial differential equations. In this seminar, I will share my teaching experience at the Faculty of Agriculture, Kyoto University, before turning to recent advances in research at a graduate level. The principal governing equations in open channel hydraulics are the shallow water equations, which are derived from the conservation laws of mass and momentum. There are well-established results indicating the possible occurrence of discontinuities in their solutions, known in engineering as hydraulic jumps, bores, and surges. However, roll waves—propagating with discontinuities over steep channels—appear to require further mathematical investigation, as they involve well-posedness issues in initial value problems. Insightful real-world observations and computational results have emerged from practical educational courses.