I will survey results on stochastic interacting particle systems (such as TASEP, the Totally Asymmetric Simple Exclusion Process) in the presence of inhomogeneity. That is, we consider particles with variable speeds, or the space in which the system evolves contains “swamps and highways”, where particles move with variable speed. Inhomogeneity inserts multiple parameters into the system, but in some miraculous cases the system stays exactly solvable in their presence. This allows to observe interesting asymptotic phase transitions. Moreover, permutations of the multiple parameters can sometimes be realized as Markov operators on the state of the system, leading to intriguing structural properties.