He started trading at the age of 18, and rose to become the general manager of G. B. Ollivant in Ghana. In 1937, he left the firm after it was acquired by the United African Company. Leventis then formed his own company and started out as a produce buyer, partly financed by some British cotton manufacturers.
In the late 1940s, the UK authorities imposed a country by country quota on cotton imported from Africa, which was intended to influence textile and oil seed production in West Africa. The situation created the impetus for A.G. Leventis to establish a branch in Nigeria, called A.G. Leventis Nigeria Plc. Within a few years the company expanded its business line from cotton exports to merchandise trading. By the 1960s his firm had grown to become one of the largest distributors in Nigeria and one of the largest merchandise traders in the West African region. In Nigeria, he re-structured the business from general trading into a specialized trading firm and established various department stores. During this period, he thrived as a result of the nation's relatively open economy, as it was not until the 1970s that economic nationalism became a dominant initiative. His marketing style made the Leventis name familiar to many customers in Nigeria.
After his death in 1978, the A.G. Leventis Foundation was established in 1979. It has a primary focus on the cultural heritage of Cyprus and Greece, especially reflected in its collections of Cypriot antiquities displayed in several museums around the world, its restoration of cultural monuments, and its sponsorship of scholarships for postgraduate work in several fields including archeology and agriculture. The foundation also sponsors work in the areas of environmental protection and medical research.
In 1987 the A.G. Leventis Gallery was opened in the British Museum, to display Cypriot antiquities from the early Bronze Age to the Roman era. In 1997, a similar display was opened at the Fitzwilliam Museum, and in 2000 another at the Metropolitan Museum of Art, New York.
In 2008 the A.G. Leventis Foundation endowed a new chair at Cambridge University in Greek culture—the A. G. Leventis Professor of Greek Culture.
The earliest records of his family go back to the 18th Century when a young ancestor had travelled to the Peloponnese to join the abortive 1770 uprising against Ottoman rule (known to history as the ‘Orlov’ rebellion). By the time of his birth, his family counted a number of distinguished clerics who served at the highest levels — metropolitan bishops, patriarchs from the Orthodox churches of Jerusalem and Antioch, as well as from the Church of Cyprus.
Born in December 1902 in Lemythou, Cyprus to Neoklis/Papaneoklis Leventis and Salome Theocharides, he had several brothers and sisters: Costas; George, who firstly went to Egypt; Kalliopi, mother of film producer Alki David; Evagoras; Katerina, who wed Mr Myrianthousis; Charalambos; Anastasios; Christodoulos; and David.
He married Fotini, with whom he had a child, Papadopoulos Leventis.
He is also uncle to Constantine (Dino) Leventis (1938–2002), businessman and major art benefactor of Hellenic art at the Metropolitan Museum of Art in New York and other museums around the world: in April 2000, the Metropolitan Museum opened the A.G. Leventis Foundation Gallery where the museum's collection of Cypriot sculpture, terracottas, vases, jewelry and coins from the fifth and fourth centuries BC are displayed. The foundation also helped pay for similar galleries devoted to ancient Greek and Cypriot art in the British Museum, the Louvre in Paris and the Fitzwilliam Museum at Cambridge University, among others, according to its latest annual report. In Greece, it has financed extensive archeological work.