In recent years computational simulations have rapidly evolved in complexity (high order discretizations, spatial adaptivity, additional physical processes), placing ever larger strains on the time integration methods on which they rely. High spatial order necessitates comparably high order time integration. Spatial adaptivity and multiphysics processes give rise to subsets of the solution that evolve at differing time scales, or to simulations that combine nonstiff but highly nonlinear processes with others that may be highly stiff but that are frequently linear. In this talk, I will discuss recent work on advanced time integration methods (both mathematical derivation and implementation in high-quality open-source software), that allow extreme flexibility in the use of different techniques to distinct physical processes, while still allowing high orders of accuracy. I will primarily focus on the newly-developed IMEX-MRI-GARK methods (Chinomona and R., 2021), and their implementation in the ARKODE library within SUNDIALS, but I will additionally point out other recent related work.