The town was named after Norwich, Connecticut, the state from which the first settlers originated. In the original charter, the name was spelled "Norwhich", but the additional 'h' was dropped shortly after the town was chartered. Originally pronounced "Norritch" (similar to the English pronunciation of the city of Norwich) the town name has in more recent times become more commonly pronounced "Nor-wich". Norwich University was located here originally. It moved to Northfield, Vermont, in 1866 after a fire devastated the campus.
The town of Norwich was chartered on July 4, 1761, by Gov. Benning Wentworth of New Hampshire, to a group of men from Tolland County, Connecticut. Some of the original proprietors visited the area during the summer of 1763 and several summers after in order to survey, clear, and prepare the lands for settlement. The first families were living there by 1767. An illustrative description of the early years of the town is found in the biography of John Slafter. At that time, both Colonies of New York and New Hampshire lay claim to what is today the State of Vermont, and both governors were selling land charters in the disputed area. A useful history of this situation is given in the Wikipedia article on New Hampshire grants.