Dunsink's most famous director was William Rowan Hamilton, who, amongst other things, discovered quaternions, the first non-commutative algebra form, while walking from the observatory to the city with his wife. The annual Hamilton Walk that commemorates this discovery begins at the observatory. He is also renowned for his Hamiltonian formulation of dynamics.
The observatory was established by an endowment of £3,000 in the will of Francis Andrews, who was Provost of Trinity College Dublin at his death on 18 June 1774. The site was established on the south slope of a low hill in the townland of Dunsink, 84m above sea level.  The South Telescope, a 12-inch Grubb instrument, is a refracting (i.e. it uses lens) telescope built by Thomas Grubb of Dublin, and completed in 1868.  The achromatic lens, with an aperture of 11.75 inches, was donated by Sir James South in 1862, who had purchased the lens from Cauchoix of Paris 30 years earlier.  He had intended it for a large but troubled equatorial that came to fruition in the 1830s, but was dismantled around 1838.   (See also Great refractors)
The entry for the observatory in Thom's Directory (1850) gives the following account of the observatory,
::ASTRONOMICAL OBSERVATORY OF THE UNIVERSITY OF DUBLIN, DUNSINK
- Astronomer Royal, Sir William Rowan Hamilton, A.M., LL.D.
- Assistant Astronomer, Charles Thomson, esq.
This Observatory, endowed by Francis Andrews, esq., LL.D., Provost of Trinity College, and erected in 1785, was placed, by statute, in 1791, under the management of the "Royal Astronomer of Ireland," an appointment first filled by Dr. Henry Ussher, and subsequently by Dr. Brinkley, Bishop of Cloyne.
The Institution is amply furnished with astronomical instruments, and is open to all persons interested in astronomical science, on introduction to the resident Assistant. It is situated in Lat. 53° 23' 13" N., Long. 6° 20' 15" W. 
Dublin Mean Time, the official time in Ireland from 1880, was the mean solar time at Dunsink, just as Greenwich Mean Time (GMT) was the mean solar time at Greenwich Royal Observatory near London.  In 1916, Ireland moved to GMT. In 1936, Trinity College stopped maintaining the observatory and rented out the land.
Éamon de Valera, who had driven the establishment of the DIAS in 1940, added a School of Cosmic Physics to it in 1947, partly in order to revive the observatory, for which it was given responsibility.
The named chair Andrews Professorship of Astronomy was associated with the directorship of Dunsink Observatory during the time that the observatory was part of Trinity College Dublin (TCD).
The observatory is currently part of the Dublin Institute for Advanced Studies (DIAS). It provides accommodation for visiting scientists and is also used for conferences and public outreach events. Public talks on astronomy and astrophysics are given regularly at the observatory by professional and amateur astronomers. Stargazing events are also held using the Grubb telescope.
|1783–1790||Rev. Henry Ussher||Andrews Professor of Astronomy||Died in office|
|1792–1827||Rev. John Brinkley||Andrews Professor of Astronomy, Royal Astronomer of Ireland (from 1793)||Appointed Bishop of Cloyne in 1826|
|1827–1865||Sir William Rowan Hamilton||Andrews Professor of Astronomy, Royal Astronomer of Ireland||Appointed as a 21-year-old undergraduate. In addition to astronomy, he worked on mathematics. He developed what is now known as Hamiltonian mechanics, and the system of quaternions, having discovered them in 1843. He died in office.|
|1865–1874||Franz Brünnow||Andrews Professor of Astronomy, Royal Astronomer of Ireland||Retired due to failing health and eyesight|
|1874–1892||Sir Robert Stawell Ball||Andrews Professor of Astronomy, Royal Astronomer of Ireland||In 1892 became Lowndean Professor of Astronomy and Geometry at Cambridge|
|1892–1897||Arthur Alcock Rambaut||Andrews Professor of Astronomy, Royal Astronomer of Ireland||In 1897 became Radcliffe Observer at Oxford|
|1897–1906||Charles Jasper Joly||Andrews Professor of Astronomy, Royal Astronomer of Ireland||Died in office|
|1906–1912||Sir Edmund Taylor Whittaker||Andrews Professor of Astronomy, Royal Astronomer of Ireland||In 1911 became a professor at Edinburgh|
|1912–1921||Henry Crozier Keating Plummer||Andrews Professor of Astronomy, Royal Astronomer of Ireland||In 1921 became professor of mathematics at the Artillery College in Woolwich|
|1921–1936||Charles Martin||Acting Director, assisted by F J O'Connor (1908-1987). Died in office|
|1936–1947||Vacant||No astronomical work was done|
|1947–1957||Hermann Alexander Brück||Director of DIAS School of Cosmic Physics||In 1957 became Astronomer Royal for Scotland|
|1958–1963||Mervyn Archdall Ellison||Director of DIAS School of Cosmic Physics||Died in office|
|1964–1992||Patrick Arthur Wayman||Andrews Professor of Astronomy (from 1984, honorary), Director of DIAS School of Cosmic Physics||Retired, with a short gap before the next appointment.|
|1994–2007||Evert Meurs||Senior Professor DIAS||Retired|
|2007–2018||Luke Drury||Andrews Professor of Astronomy (honorary), Director of DIAS School of Cosmic Physics||Retired|
|2018–present||Peter T. Gallagher||Senior Professor and Head of Astronomy and Astrophysics, DIAS|