When is the Moon at last quarter?

The Moon always has half its sphere lit by the Sun, but from Earth we see a varying fraction of that half. At last quarter, half of the visible disk appears bright, which is actually half of the illuminated half: a quarter of the Moon. Hence the name, and why in English it's called the last quarter or third quarter. It shouldn't be confused with the first quarter: both phases show half a disk, but at last quarter the illuminated side is the opposite one, and the Moon appears in the early morning hours instead of at dusk.

The full cycle of phases repeats roughly every 29.5 days, so the last quarter returns once per lunation, always after the full moon and before the new moon. Its most distinctive feature is timing: it rises around midnight and sets at midday. It is, therefore, the phase for early risers; at dawn it's high in the sky and remains clearly visible against the blue daytime sky through the early morning. From the northern hemisphere, the illuminated edge is the one facing the Sun, so at that hour it points east.

The date changes every month and doesn't match across all calendars because the phase is defined by a specific instant, the same for the whole planet, that each region translates into its local time: what happens at dawn in Europe falls on the previous afternoon in the Americas. For a specific date it's best to check an ephemeris or an observatory's almanac, not a mental rule. What is invariable is the order of phases: full, waning gibbous, last quarter, thin early-morning waning crescent, and new. If one night the Moon rises late and sits halfway up the sky at dawn, split cleanly down the middle, it's at last quarter.