Why isn't the Moon out?

The question hides a false assumption: that the Moon should be in the night sky every night. It doesn't work that way. As it moves along its orbit, the time at which it appears over the horizon keeps shifting. The full cycle of phases repeats roughly every 29.5 days, while one orbital revolution around Earth takes 27.322 days; the difference exists because, during that time, Earth has also moved along its path around the Sun.

Each phase has its own schedule. The full moon rises at sunset and sets at dawn: it's the only one that accompanies the entire night. The first quarter rises at midday and sets around the middle of the night, so by three in the morning it's no longer there. The waning gibbous rises well into the night and sets in the morning. The last quarter rises in the middle of the night and sets at midday. The thinnest waning crescent appears shortly before dawn and sets in the afternoon. If you go out to look at ten at night and find nothing, it's most likely that the Moon hasn't risen yet or has already set hours ago.

The extreme case is the new moon. Then the illuminated face points toward the Sun and the night face is oriented toward us, so there is nothing to see. It also coincides with the Sun in the sky: it rises and sets with it and is washed out by daylight. During the new moon and the thinnest crescent phases, all or nearly all of the disk is in shadow. Finally, it's worth remembering that the Moon is also frequently visible during the day: it isn't an exclusively nocturnal body, and much of its nighttime absence is made up for by daytime appearances.