This paper presents a broad, pathfinding design space exploration of memory management units (MMUs) for heterogeneous systems. We consider a variety of designs, ranging from accelerators tightly coupled with CPUs (and using their MMUs) to fully independent accelerators that have their own MMUs. We find that regardless of the CPU-accelerator communication, accelerators should not rely on the CPU MMU for any aspect of address translation, and instead must have its own, local, fully-fledged MMU. That MMU, however, can and should be as application-specific as the accelerator itself, as our data indicates that even a 100% hit rate in a small, standard L1 Translation Lookaside Buffer (TLB) presents a substantial accelerator performance overhead. Furthermore, we isolate the benefits of individual MMU components (e.g., TLBs versus page table walkers) and discover that their relative performance, area, and energy are workload dependent, with their interplay resulting in different area-optimal and energy-optimal configurations.