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His academic career began with a research fellowship at Clare Hall, Cambridge, after which he went on to teach art history at Manchester University. While still in his early thirties, Penny was appointed to the Slade Professorship at Oxford University and to a senior research fellowship at King's College, Cambridge. He was the co-author, with Francis Haskell, of Taste and the Antique, a study of the formation of the canon of classical sculpture published in 1984.
Between 1984 and 1989 Penny was keeper of the department of Western art at the Ashmolean Museum, Oxford and professorial fellow of Balliol College, Oxford. In 1990 he began a long association with the National Gallery, joining the institution as Clore Curator of Renaissance Painting. Shortly afterwards, in 1991, he identified the Madonna of the Pinks belonging to the Duke of Northumberland as a genuine Raphael, and not a copy of a lost original as was previously supposed. The painting came to public prominence in 2002 when the Gallery undertook a major fundraising campaign in order to prevent the painting's sale to the Getty Center in Los Angeles. Earlier that year Penny made an unsuccessful bid for the directorship of the National Gallery, the post going to Charles Saumarez Smith. Again in 2002, Penny was appointed senior curator of sculpture at the National Gallery of Art in Washington, D.C. Following Saumarez Smith's early resignation from his post, Penny was once again a candidate for heading the London National Gallery, and this time he succeeded. He was appointed a Knight Bachelor in the Queen's 2015 Birthday Honours.