We consider partial differential equations that describe the conservation of one or several quantities, possibly taking an additional dissipation mechanism into account, set on the real line. Such models are for instance relevant in gas dynamics or in the study of road traffic. When the initial data of these conservation laws are monotonic and bounded, a probabilistic theory can be developed by interpreting the solutions as cumulative distribution functions on the line. The study of the associated stochastic processes and their approximations by interacting particle systems provides a Lagrangian description of the solution, that will be used to derive existence results together with numerical schemes, as well as estimates of stability and convergence to equilibrium or traveling waves.