Tashlich — "throwing or casting away" — is a custom which involves going to a body of water on the first afternoon of Rosh Hashanah and reciting the Tashlich prayer (see The ArtScroll Siddur, p. 770). If the first day is Shabbat Tashlich is postponed until the following day. If necessary, Tashlich may be done until Hoshanah Rabah.

1. Michah 7:19
   He will again be merciful unto us; He will suppress our iniquities; And cast into the depths of the sea all their sins.
2. Shmot 34:7
   Abundant in kindness, abundant in truth; preserver of kindness for thousands of generations.
3. Hagaon Rav Moshe Shapiro, shlita
   Action, thought and speech of humans have spiritual impact because we are created in the image of God. Spiritual effects are beyond the limitations of time-space. We ask God in His mercy to be uneven in His consideration of good and evil actions. Our mitzvot should be "preserved for thousands of generations" and our aveirot should be "cast into the depths of the sea."
4. Zohar Parshat Naso 131
   When Avraham was taking Yitzchak to the Akeida the Satan created a deep, fast flowing river in the path of Avraham. Avraham waded into the river up to his neck and forded the stream in order to do the will of God.
   [Throughout the centuries the Jewish people have repeated the self-sacrifice of Avraham and Yitzchak and have braved fire and water to do the will of our Creator. We ask God to remember and count that merit of the Jewish people.]
5. Maharam Yaffe
   The fish in the river remind us that human life is as frail as a fish in the sea that can at any moment be caught in the fisherman’s net.
6. Babylonian Talmud, Avodah Zarah 3b
   Rav Yehudah said in the name of Shmuel, "What is the meaning of the verse, ‘And He made people like the fish of the sea?’ Just as the fish of the sea die as soon as they leave the water so also when people separate themselves from the Torah and the mitzvot they die."