Webber, the son of a Swiss sculptor, was born in London but was sent abroad for his early artistic education. In Berne he studied under J.L. Aberli, the originator of a picturesque manner of portraying Swiss mountain scenery. Returning to London, Webber studied at the Royal Academy schools. In 1776, through the influence of Dr. Solander, he was appointed official draughtsman on Captain Cook's third voyage. Cook himself emphasized the importance of professional drawings as a complement to scientific records: "Mr. Webber was engaged to embark with me," he wrote, "for the express purpose of supplying the unavoidable imperfections of written accounts." Webber made numerous drawings not only of landscape but also, in Cook's words, "of everything that was curious, both within and without doors." His sketches of natives, their houses, clothing, utensils, and customs add a vivid gloss to Cook's narrative. Webber was an eyewitness of Cook's death, and his painting of it, engraved by Byrne and Bartolozzi, became the standard representation of that tragic event.
On his return to London, Webber superintended the engraving of his drawings for the official account of the expedition published by the Admiralty in 1784, although his sketches of native faces and costume inevitably suffered some "Europeanization" in the engravers' hands. Webber himself engraved and published a series of Views in the South Seas (1787-92), and exhibited at the Royal Academy several paintings based on the voyage. He was elected Associate of the Royal Academy in 1785, and Royal Academician in 1791. From 1790 his exhibits there included English landscapes. He made several sketching tours in Wales and Derbyshire, delineating the rocky terrain with the same studious, attentive eye that he had bent on less familiar regions. He appears never to have essayed imaginative or deliberately romantic subjects. His draughtsmanship is invariably fine and detailed. His coloring is restrained and delicate: the contemporary comment of Edward Edwards that he is "frequently too gaudy" is difficult to understand.
Webber's particular friends among artists were Farington and Hearne, to each of whom he bequeathed drawings. He died in London on 29 May 1793. (John Baskett and Dudley Snelgrove, English Drawings and Watercolors 1550-1850 in the Collection of Mr. and Mrs. Paul Mellon, exh. cat. The Pierpont Morgan Library, New York, 1972: 44)