Can I run exposed Romex wire in my garage?

It depends on how exposed it is and your local interpretation of the National Electrical Code, but generally you cannot run Romex exposed on the surface of walls or ceilings in a garage where it is subject to physical damage. The NEC Article 334 states that NM-B cable must be protected from physical damage where necessary and cannot be run exposed in unfinished areas like garages if it is run along the surface where it could be snagged by tools, lumber, or vehicles. If your garage walls are finished with drywall and the Romex is fished inside the walls or stapled inside stud bays and then covered, that is allowed. If your garage has open studs and you staple Romex along the face of studs or across joists below 8 feet, most inspectors will fail it. The approved methods for open garage wiring are to either protect Romex by running it through holes drilled in the center of studs and joists, install running boards, use conduit or MC cable for the exposed portions, or mount plywood sheathing to protect the wiring. In many jurisdictions, garages are considered to require protection, so exposed Romex must be in conduit, typically EMT or PVC, or you must use a more robust cable type. Short exposed runs from a junction box to a fixture in the ceiling joist area are often allowed if closely supported and not subject to damage, but the safest code-compliant approach for an unfinished garage is to use THHN in conduit or metal-clad cable for all exposed work and reserve Romex for inside finished walls.