Applications of density functional theory (DFT) to computational chemistry and solid-state physics rely on a “Jacob's Ladder” of progressively more complicated approximations to the many-body exchange-correlation (XC) density functional. Accurate, computationally tractable DFT calculations on large and periodic systems remain challenging for existing XC functionals. Simple XC functionals on the three lowest rungs of Jacob's Ladder are insufficiently accurate for many properties, while fourth-rung hybrid functionals incorporating nonlocal information can be prohibitively expensive. This perspective presents our work toward a compromise, a new class of “Rung 3.5” functionals that incorporate a linear dependence on the nonlocal one-particle density matrix. This work reviews these functionals' formal underpinning, numerical performance, and prospects for modeling solids and surfaces. © 2012 Wiley Periodicals, Inc.