He was born on 11 May 1932 in Breslau, Germany. Starting in 1949, he studied mathematics, physics, and philosophy at the Georg-August-University at Göttingen, where he received his doctorate in 1954. Until 1956 he worked in the Mathematical Research Institute of Oberwolfach. In 1958 he became a lecturer at the Technical University of Munich, where he obtained his Habilitation. In 1958 he moved to the Technical University of Darmstadt, where in 1962 he became a professor, and remained until his retirement. From 1976 to 1984 he was a visiting professor at Caltech.
Laugwitz worked in differential geometry of infinite dimensional vector spaces (his dissertation) and in Finsler geometry. In 1958 he and Curt Schmieden developed their own approach to infinitesimals through field extensions, independently of Abraham Robinson. They described this as "infinitesimal mathematics" and leading back to the historical roots in Leibniz. In 1996 he published the standard biography of Bernhard Riemann.