A recent trend toward wider tires (cf Heine, "Performance of Tires") also led to wider rims and a closer match of rim and tire widths.
In the past it was considered that tires need to be considerably wider than their rims. For example, Sheldon Brown requires at least 40 mm nominal width in a 23 mm rim (measure between inner flanges). ISO 4210 (cf Mavic) requires at least a 37 mm tire on the same rim.
As a recent example, my new bike (2017 Merida CX 600) came with a 700x23C rim and 32-622 tires. Suggesting that is nonetheless a safe combination. The rim lacks hooks though (see below), which might make it comparable to a narrower specification.
New developments, such as tubeless, hook-less rims, and perhaps in tires beads might change this into a more complex question. Perhaps it even allows closer matched widths to be used safely.
What is the smallest width of a tire for a given rim width? (It might be convenient to consider different cases and technologies in separate answers.)