Buck was born in Castle Street, Cork. Becoming an accomplished miniaturist in the 1780s whilst still in Ireland, he made a permanent move to London in 1795 - his residences there included 174 Piccadilly (1795–8), Frith Street, Soho (1799–1802) and Bentinck Street (1813–20). His patrons included Angelica Catalani (an opera singer), JP Kemble, Sir Francis Burdett, Thomas Hope, George IV, the duke of York and his mistress Mary Anne Clarke. A major influence on Regency culture (producing plates of contemporary costume as well as genre pictures of family and classical scenes and illustrations for Laurence Sterne's Sentimental Journey), he was himself much influenced by the Greek Revival (the furniture, vases - which he collected -, sculptures, costumes and even hairstyles in his works are all ancient Greek). He also acted as a painting teacher, as well as exhibiting more than 170 miniatures and small full-length portraits at the Royal Academy between 1795 and 1833. He died at 15 Upper Seymour Street, London.
Proposals for publishing by subscription 100 engravings from paintings on Greek vases which have never been published, drawn and etched by Adam Buck from private collections now in England, dedicated to the earl of Carlisle (1811) - intended as a continuation of Sir William Hamilton's Collection of Engravings from Ancient Vases … (1791–7)