Rabbi Abba was one of the great Amoraim of the third generation, between the late third and early fourth centuries. Born in Babylonia, he studied under Rav Huna, head of the Sura yeshiva, and Rav Yehuda, head of the Pumbedita yeshiva; sources also list him among the students of Rav and Shmuel.

Despite the opposition of his teacher Rav Yehuda, he went up to the Land of Israel. The Talmud relates that Rabbi Abba would kiss the rocks of Akko (Ketubot 112a), a sign of his deep love for the Land. Before arriving, he prayed that his words would be accepted by its sages (Beitzah 38a). According to Hevrat Pinto, he merited to see Rabbi Yochanan and Reish Lakish.

He settled in Eretz Yisrael and gained a high reputation: in the Babylonian academies, "our teachers in the Land of Israel" refers to Rabbi Abba (Sanhedrin 17b). Rav Safra was among his students. According to Hevrat Pinto, in his later years he served as av beit din alongside Rabbi Ami, the rosh yeshiva, and followed him to Caesarea.

A wealthy silk merchant, he gave charity in secret: he would tie money to the back of his garment, and the poor would untie the knot and take it (Ketubot 67b). The Talmud also tells of a plot of land he bought unaware that Rav Giddel had been negotiating for it; since neither wished to use it, it became known as "the field of the Rabbis" (Kiddushin 59a).

His teachings in halacha and aggada appear throughout both Talmuds and the Midrashim, and a complex discussion in monetary law bears his name, "Naskha de-Rabbi Abba" (Bava Batra 33b). He died at a very advanced age, apparently in the Land of Israel.