Nonlinearly Partitioned Runge--Kutta (NPRK) methods are a newly proposed class of time integration schemes which target differential equations in which different scales, stiffnesses or physics are coupled in a nonlinear way. In this talk, I will provide a broad overview of this new class of methods. First, I will motivate these methods as a nonlinear generalization of classical Runge--Kutta (RK) and Additive Runge--Kutta (ARK) methods. Subsequently, I will discuss order conditions for NPRK methods; we obtain the complete order conditions using an edge-colored rooted tree framework. Interestingly, NPRK methods have nonlinear order conditions which have no classical additive counterpart. We will show how these nonlinear order conditions can be used to obtain embedded estimates of state-dependent nonlinear coupling strength and present a numerical example to demonstrate these embedded estimates. I will then discuss how these methods yield efficient semi-implicit time integration of numerical partial differential equations; numerical examples from radiation hydrodynamics will be presented. Finally, I will discuss our recent work on multirate NPRK methods, which target problems with nonlinearly coupled processes occurring on different timescales. We will discuss properties of these multirate methods such as timescale coupling, stability and efficiency, and conclude with several numerical examples, such as a fast-reaction viscous Burgers’ equation and the thermal radiation diffusion equations.