[3] Morgantown was founded by Robert Bowles and Samuel Teeters, who first laid out 52 lots in March 1831.

Indiana State Road 252 passes through the town center as Washington Street, leading northwest 10 miles (16 km) to Martinsville, the county seat.

State Road 135 joins SR 252 on Washington Street but turns south out of town on Marion Street, leading 12 miles (19 km) to Nashville.

The two highways leave town together and lead northeast 6 miles (10 km) to Trafalgar.

According to the U.S. Census Bureau, Morgantown has a total area of 0.38 square miles (0.98 km2), all land.

[1] Indian Creek runs along the southern edge of the town, flowing west to the White River south of Martinsville.

31.0% of all households were made up of individuals, and 13.6% had someone living alone who was 65 years of age or older.

26.2% of all households were made up of individuals, and 13.4% had someone living alone who was 65 years of age or older.