He was co-developer and co-eponym of the Runge–Kutta method (German pronunciation: [ˈʀʊŋə ˈkʊta]), in the field of what is today known as numerical analysis.

In addition to pure mathematics, he did experimental work studying spectral lines of various elements (together with Heinrich Kayser), and was very interested in the application of this work to astronomical spectroscopy.

In 1904, on the initiative of Felix Klein he received a call to the University of Göttingen, which he accepted.

His daughter Iris also became a mathematician and his son Wilhelm was an early developer of radar.

Another of his daughters, Nerina (Nina), married the mathematician Richard Courant.