The waning crescent covers elongation from two hundred and seventy degrees back to three hundred and sixty, which is zero again, between last quarter and the next conjunction. Less than half the disc is lit and the lit limb faces the Sun, so the crescent is seen in the eastern sky before dawn.
The phases are produced by the changing angle between the Sun, the Earth and the Moon, and the quantity that defines them is elongation, the Moon's angular distance from the Sun as seen from the Earth. The complete round from one new moon to the next is the synodic month, about 29.53 days, and it is longer than the sidereal month of about 27.32 days in which the Moon returns to the same place among the stars, because the Earth has moved along its orbit in the meantime and the Moon must travel further to catch up.
The last visible crescent before conjunction is the mirror of the first visible crescent after it, and it is the harder of the two to catch because it is a morning object seen against a brightening sky. Both are limited by the same combination of elongation, altitude and atmospheric clarity.
In the thin crescent phases the dark part of the disc is often faintly visible, lit by sunlight reflected from the Earth. The effect is called earthshine, and it is bright enough to be seen with the naked eye under good conditions, which is why the whole disc can be made out within the arms of a crescent. It is the reason the old moon is described as being held in the new moon's arms in more than one tradition.