Gaon originally referred in Ancient Hebrew to arrogance and haughty pride (Amos 6:8). Later became known as pride in general: whether good or bad ('Pride [of]'; Late medieval and modern Hebrew for 'genius'). Today it may refer to:

• One of the Geonim
  Geonim
  Geonim were the presidents of the two great Babylonian, Talmudic Academies of Sura and Pumbedita, in the Abbasid Caliphate, and were the generally accepted spiritual leaders of the Jewish community world wide in the early medieval era, in contrast to the Resh Galuta who wielded secular authority...
  
  , that is to say the heads of the two major academies, at Pumbedita
  Pumbedita
  Pumbedita was the name of a city in ancient Babylonia close to the modern-day city of Fallujah....
  
   and Sura
  Sura (city)
  Sura was a city in the southern part of ancient Babylonia, located west of the Euphrates River. It was well-known for its agricultural produce, which included grapes, wheat, and barley...
  
  , and later in Baghdad, during the period 589-1040. Prominent Geonim are:
  • Sar Shalom
  • Natronai II, Gaon of Sura (Gaon to 857)
  • Amram Gaon
    Amram Gaon
    Amram Gaon was a famous Gaon or head of the Jewish Talmud Academy of Sura in the 9th century. He was the author of many Responsa, but his chief work was liturgical.He was the first to arrange a complete liturgy for the synagogue...
    
    , Gaon of Sura (Gaon 857-875)
  • Hai Gaon
    Hai Gaon
    Hai ben Sherira , was a medieval Jewish theologian, rabbi and scholar who served as Gaon of the Talmudic academy of Pumbedita during the early 11th century. He was born in 939 and died on March 28, 1038...
    
    
  • Saadia Gaon
    Saadia Gaon
    Saʻadiah ben Yosef Gaon was a prominent rabbi, Jewish philosopher, and exegete of the Geonic period.The first important rabbinic figure to write extensively in Arabic, he is considered the founder of Judeo-Arabic literature...
    
    
  • Sherira Gaon
    Sherira Gaon
    Rav Sherira Gaon was the head of the Academy of Pumbeditha. He was one of the most prominent Geonim of his period, and the father of Hai Gaon, who succeeded him as gaon.Sherira was born in 906 and died in 1006. Rav Sherira Gaon (Hebrew: רב שרירא גאון or R. Sherira ben Ḥanina Gaon, Hebrew: רב...
    
    
  • Yehudai Gaon
    Yehudai Gaon
    Yehudai ben Nahman was the head of the yeshiva in Sura from 757 to 761, during the Gaonic period of Judaism...
    
    
  • Samuel ben Hofni
    Samuel ben Hofni
    Samuel ben Hofni was the last gaon of Sura...
    
    

• An honorific title given to a few leading rabbis of other countries in the same period, such as:
  • Nissim Gaon
    Nissim Ben Jacob
    Nissim ben Jacob , was a rabbi best known today for his Talmudic commentary ha-Mafteach, by which title he is also known.-Biography:Rav Nissim studied at the Kairouan yeshiva, initially under his father - Jacob ben Nissim who...
    
    
• Specific rabbi
  Rabbi
  In Judaism, a rabbi is a teacher of Torah. This title derives from the Hebrew word רבי , meaning "My Master" , which is the way a student would address a master of Torah...
  
  s of later periods, called "gaon" (wise):
  • The Vilna Gaon
    Vilna Gaon
    Elijah ben Shlomo Zalman Kramer, known as the Vilna Gaon or Elijah of Vilna and simply by his Hebrew acronym Gra or Elijah Ben Solomon, , was a Talmudist, halachist, kabbalist, and the foremost leader of non-hasidic Jewry of the past few centuries...
    
    
  • The Rogatchover Gaon
    Rogatchover Gaon
    Joseph Rosen known as the Rogatchover Gaon, , and also often referred to by the title of his main work Tzofnath Paneach , , was a rabbi and one of the most prominent talmudic scholars of the early 20th-century, known as a genius because of...
    
    
  • The "Lisser Gaon", Rabbi Jacob of Lissa
  • The Steipler Gaon

Many great Rabbis, although not formally referred to as the "Gaon of ...", are nonetheless sometimes lauded with this honorific as a mark of respect, and as a means to indicate greatness.