Falls Township is one of the twenty-five townships of Muskingum County, Ohio, United States. According to the 2020 census, the township had a population of 8,718 residents. Much of the city of Zanesville, the county seat of Muskingum County, occupies eastern Falls Township, and the census-designated place of North Zanesville is located northeast of t…Falls Township is one of the twenty-five townships of Muskingum County, Ohio, United States. According to the 2020 census, the township had a population of 8,718 residents. Much of the city of Zanesville, the county seat of Muskingum County, occupies eastern Falls Township, and the census-designated place of North Zanesville is located northeast of the city of Zanesville.