Transport equations are used to model complex systems in a variety of applications, including mean field games. Such equations often involve a drift depending on the solution or other parts of the system in a nonlinear way. Such nonlinear transport equations can be understood by developing a theory for transport equations with irregular drifts. In this talk, I will outline the well-posedness theory for transport equations with certain classes of drifts for which the divergence has a one-sided bound, yielding contractive or expansive behavior, depending on the direction in which the equation is posed. The analysis requires studying the relationship between the transport and continuity equations and the the associated ODE flow. The theory is then used to discuss certain nonlinear transport equations arising in the study of finite state-space mean field games.