Without a formal education Arrowsmith went to London c. 1770 and, after working as a surveyor, established himself as a mapmaker and publisher. His large world map (1790) established his reputation. A second such map published in 1794 was accompanied by an explanatory volume. Other notable works included a map of North America (1796), a chart of the Pacific Ocean (1798), and his Atlas of South India (1822).

After Arrowsmith's death, the business was carried on by his sons, Aaron and Samuel, who published geography manuals and a number of atlases. From 1839 the business was conducted by his nephew, John Arrowsmith, himself an eminent cartographer, who published the London Atlas 4 vol. (1834), the best set of maps then in existence. This work was followed by a long series of elaborate and carefully executed maps embodying the results of contemporary exploration. The maps of Australia, North America, Africa, and India were especially valuable.