In 1826 Moffat's father moved the family to Edinburg, United Kingdom. In 1847 he began a business as a portrait engraver and in 1853 began a photography business. He became quite successful, and by 1887 was employing twenty assistants. His son Frank joined the firm, and his son Arthur, who had won a silver medal a: Edinburgh, evidently produced oil portraits based on his father's photographs. Frank took over the business upon Moffat's death. In addition to his photography, he was also an amateur violinist and an amateur landscape watercolorist. He took photographs of many nineteenth-century notables, including Henry Fox Talbot, Sir David Brewster, Lord Moseberry and Mr. and Mrs. Gladstone.