[1][2] His father, Jacob Greenberg, died when Hank was six and his mother, Ada Rheingold, married a dairy farmer.

[2] He received the Bronze Star,[4] and the French Legion of Honour as a result of his military service in the European theatre of World War II.

[5] Greenberg attended the University of Miami, where he was a member of Sigma Alpha Mu fraternity,[6] and received his bachelor's degree in 1948.

[12] In an interview with Reactions magazine in March 2010, serialized over three parts, Greenberg stated that he did not condone AIG's strategy of selling non-core assets to pay back the United States government, and believed the terms under which AIG was provided access to bail-out funds needed to be renegotiated.

[16] In February 2014, Greenberg led a group through Starr Investment Holdings that acquired health insurance claims processor MultiPlan Inc. for around $4.4 billion.

[14] In August 2009, the U.S. Securities and Exchange Commission charged Greenberg for his involvement in alleged fraudulent accounting transactions that inflated AIG’s finances.

The New York State trial counsel said a guilty verdict was needed to "send a message to CEOs of other companies" that "you can’t do this sort of thing.

"[20] The case was resolved on February 10, 2017, subject to a settlement in which Greenberg admitted to fraud and agreed to pay $9 million.

[25] In November 2012, a Manhattan court dismissed Greenberg's claims that the Federal Reserve Bank of New York breached its fiduciary duties to AIG shareholders.

[33] He is a member of the US-China Business Council, honorary chair of the U.S.-China Policy Foundation[34] and is vice-chairman of the board of directors of the National Committee on United States – China Relations.

[35] The Maurice "Hank" Greenberg Scholarship, administered in his name by the US-China Education Trust, supports the studies of ten Chinese students from low-income families each year at Yunnan University.

[14] In July 2022, Greenberg announced the founding of a group composed of senior U.S. business and policy leaders who share the view that the United States should engage more constructively with China.

[42] He serves as a member of the board of overseers of the Weill Cornell Medical College of Cornell University, as a life trustee of New York University, a trustee for the School of Risk Management, Insurance, and Actuarial Science and is the chairman of the Academic Medicine Development Company (AMDeC) Foundation.

[45] He is on the board of directors of the International Rescue Committee, is a former trustee of the American Museum of Natural History[46] and is active in a number of other civic and charitable organizations.

As chairman of The Starr Foundation, Greenberg oversees the disbursement of major financial support to academic, medical, cultural and public policy institutions.

According to David E. Sanger, Chief Washington Correspondent for The New York Times,[48] Greenberg listened for fifteen minutes while Ahmadinejad continued talking about the Palestinians, World War II, and if the Holocaust killings had happened at all.