He is best known for his co-discovery (with Johann Benedict Listing) of the Möbius strip, a non-orientable two-dimensional surface with only one side when immersed in three-dimensional Euclidean space. He was the first to introduce homogeneous coordinates into projective geometry. Möbius transformations, important in projective geometry, and not to be confused with the Möbius transform of number theory, also bear his name. His interest was also in number theory, and the important Möbius arithmetical function μ(n) and the Möbius inversion formula are named after him.
He was a descendant of Martin Luther.