Integrable stochastic particle systems in one space dimension, like the Totally Asymmetric Simple Exclusion Process (TASEP), have been studied for over 50 years (introduced simultaneously in biology and mathematics in 1969-70). They strike a balance between being simple enough to be mathematically tractable and complicated enough to describe many interesting phenomena. Many natural questions about these systems can be generalized by introducing multiple parameters. The interplay between these parameters is powered by the Yang-Baxter equation, which brings new intriguing results to the well-traveled territory. In particular, I will discuss new Lax-type equations for the Markov semigroups of the TASEP and its relatives. Based on a joint work with Axel Saenz.