Stoddart is best known for his civic monuments, including 10-foot (3.0 m) bronze statues of David Hume and Adam Smith, philosophers during the Scottish Enlightenment, on the Royal Mile in Edinburgh, and others of James Clerk Maxwell, William Henry Playfair and John Witherspoon.
[2] He was born in Edinburgh, though his father, also an artist, moved the family to the village of Elderslie in Renfrewshire, where the young Stoddart immediately noticed the monument there at William Wallace's purported birthplace.
[2] Stoddart has recalled an epiphany moment several times: when, after finishing a riveted metal pop-art sculpture (praised by his tutors) he found a bust of the Apollo Belvedere, "I thought my pop-riveted thing was rubbish by comparison.
[2] He has repeatedly criticised winners of the Turner Prize, such as Damien Hirst — "there's plenty of them" — and Tracey Emin, whom he calls "the high priestess of societal decline".
[2] Stoddart has characterised modern art as dominated by left-wing politics, to the extent that "certain artistic forms likewise became suspect: the tune; the rhyme; the moulding; the plinth" as coercive and overly traditional.
[3] Stoddart developed his theme on the quietism of monumental art and its relation to Schopenhaurian resignation in a lecture to the Wagner Society of Scotland on 2 March 2008.
[9] Despite their idiomatic differences, Raymond McKenzie argues that the works of both Ian Hamilton Finlay and Stoddart combine formal and intellectual elegance with sharp, sometimes satirical critiques of contemporary society.
"And yet, after having said all this about Modernism, I consider myself a Modernist – but in the context of a vast application of the term extending miles beyond the pokey wee official area to which usually it is confined.
[14] Stoddart's statue of James Clerk Maxwell, a physicist, stands in George Street in Edinburgh and a memorial to Robert Louis Stevenson, a novelist, is on Corstorphine Road.
Italia, a 2.6 metre, glass re-in-forced polymer statue on top of Ingram Street represents the contribution of Italian traders to the area.
Classical in style, the female form is swathed in a chiton and carries symbols of ancient Italy: a palm branch in her right hand and an inverted cornucopia in her left.
Opposite, on a plinth on the street, stands Mercurial, cast in bronze and with the adjectival form of the name, it complements the duality of the other two with an underlying unity.
[17] Putative projects include a monument to Willie Gallacher, the Paisley-born Communist MP, championed by Tony Benn and funded by a public appeal and "Oscar", an amphitheatre carved into the rock on the Scottish coast dedicated to Ossian, the mythical Scot bard.
[3][18] In 2019 Stoddart made a 14-foot-tall (4.3 m) statue of Leon Battista Alberti for Walsh Family Hall of Architecture of the University of Notre Dame, in the United States, his single tallest work.
[19] During 2000 to 2002 the Queen's Gallery at Buckingham Palace was renovated in the neo-classical style under the direction of John Simpson, envisioned as "building visible history".
[22] Stoddart has also worked on busts of living figures whom he admires, often fellow-classicists including philosopher Roger Scruton, architects Robert Adam and John Simpson, architectural historian David Watkin, and politician Tony Benn.