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Robert was the son of Charles Goldsborough, and was born on the family estate (Horns Point) at Cambridge in Dorchester County, Maryland. His father owned over 10,000 acres (40 km²) of land, and became a member of the Governor’s Council in 1760. As a young man Robert travelled to London to study law at the Inner Temple, where he was admitted to the bar in 1754. He remained there practicing law as a Barrister until 1759 when he returned to America. He attended the Philadelphia College (now the University of Pennsylvania), and graduated in 1760 before resuming the practice of law in Cambridge, Maryland.
Goldsborough was the sheriff of Dorchester County from 1761 to 1765. He began his service in the Maryland Assembly in 1765, and served as the colony’s Attorney General in 1766. He became active in the protests that led to the American Revolution, joining Maryland’s "Committee of Safety". In 1774 he was sent as a delegate to the Continental Congress, where he served until May 1776. Later that year he attended the Convention that drafted Maryland’s first state constitution. Under the new state constitution, he was elected to the first Maryland Senate in 1777.
Robert narrowly missed two chances to be listed formally as one of the Founding Fathers of the United States. He withdrew from the Continental Congress six weeks before the Declaration of Independence to return home and work toward’s Maryland’s first constitution. In 1787, he was selected as a representative to attend the U.S. Constitutional Convention, but didn’t attend due to poor health. He died the following year at home at Horns Point, and is buried in the Christ Episcopal Church Cemetery in Cambridge, Maryland.