The Japanese tradition of Tsukimi involves viewing the full harvest moon while eating rice dumplings and other round things. Instead of a man in the moon, folklore here sees a rabbit pounding mochi. Hence these Tsukimi dango (moon-viewing dumplings) filled with sweet bean paste and what tasted like a yuzu glaze. McDonald's even celebrates by putting a fried egg on a sandwich called the Tsukimi burger or 月見バーガー.
Another big symbol of Tsukimi is susuki grass, which explains the iPhone emoji with swaying grass and a pyramid of mochi cakes under the moon.
Tsukimi is celebrated on the 15th day of the eighth month of the lunar calendar. It's Sept. 8, 2104, Sept. 27, 2015, and Sept. 15, 2016.