Smith was probably born at Lea near Gainsborough, the son of the rector of Gate Burton, Lincolnshire. After attending Queen Elizabeth's Grammar School, Gainsborough (now Queen Elizabeth's High School, Gainsborough) he entered Trinity College, Cambridge, in 1708, and becoming minor fellow in 1714, major fellow in 1715 and senior fellow in 1739, was chosen Master in 1742, in succession to Richard Bentley. From 1716 to 1760 he was Plumian Professor of Astronomy, and he died in the Master's Lodge at Trinity.
Besides editing two works by his cousin, Roger Cotes, who was his predecessor in the Plumian chair, he published A Compleat System of Opticks in 1738, which gained him the sobriquet of Old Focus, and Harmonics, or the Philosophy of Musical Sounds in 1749.
In his will Robert Smith left £3500 South Sea stock to the University of Cambridge. The net income on the fund is annually divided equally between the Smith's Prize and the stipend of the Plumian Professor.