Around 200 C.E., Rabbi Judah Ha’Nasi (Judah the Prince) completed his great work of Jewish law, the Mishnah. Although Jewish tradition, for close to 1,500 years, forbade the writing of the oral code, Rabbi Judah decided that oral tradition was in danger of being lost due to fierce Roman persecution. This first written compilation of Jewish oral law was also a reaction to the growing diaspora in which Jews lived farther and farther apart, and knowledge of the oral law was rapidly fading.
Over the next three hundred years, this great compilation was studied, analyzed and discussed in the great learning academies that arose both in Israel and in Babylon, where there was a very large Jewish community. The great sages of this era were known as Amoraim, because they "said" or "told over" the teachings of the Oral law. (As opposed to their predecessors, the Tannaim, who were direct transmitters of the uncodified oral tradition.)
In the fifth century, two of the great Amoraim, Rav Ashi and Ravina I, realized that the discussions of these Babylonian scholars would be lost if it were not written down. They therefore began the redaction known today as the Babylonian Talmud (Talmud Bavli). In order to create the Talmud, the sages had to collect all the known discussions and legal rulings from the previous 200-300 years (known as the Gemara).
While neither Rav Ashi nor Ravina I lived to see their work completed, it was on the 13th of Kislev 4236 (475 C.E.) with the death of the last great Amora, Ravina ben Huna (known as Ravina II), the nephew of Ravina I, that the redaction of the Talmud was “closed” (meaning that, in general, nothing further was added*).
*There were some brief comments by the Saboraic rabbis a century later, but that is a topic for another day.