He studied first at Kiel University which he entered in 1900 but after one semester he moved to Berlin University where he was to study for five years. At Berlin his thesis was directed by Hermann Schwarz and his additional examiner for the oral on his thesis was Friedrich Schottky who had been appointed to Berlin in 1902 while Koebe was in the middle of his studies. Between 1904 and 1905 Koebe studied at the Charlottenburg Technische Hochschule, then he undertook research at Göttingen for his habilitation presenting his thesis in 1907.
Koebe was appointed to Leipzig University in 1910 as an extraordinary professor of mathematics. He became an ordinary professor in 1914 when he accepted a position at Jena university. He returned to Leipzig, this time as an ordinary professor, in 1926.
Koebe's work was all on complex functions, his most important results being on the uniformisation of Riemann surfaces. Shortly after 1900 Koebe established the general principle of uniformisation which had been originally conceived by Klein and Poincaré. Koebe's proof of the uniformisation theorem has been described as:-
... arguably one of the great theorems of the century.The article  describes his contributions in some detail and gives a list of 68 publications by Koebe. These are not, however, a collection of great works on a par with his proof of the uniformisation theorem. Koebe's style was pompous and chaotic and Koebe anecdotes were famous in Germany between the two wars. He did make other important contributions, however, and his circle domain conjecture is still being attacked. A special case was proved in 1993 by Z-X He and O Schramm.
Freudenthal writes in :-
He tended to deal broadly with special cases of a general theory by a variety of methods ...Freudenthal also tells us that Koebe's life-style was, as his mathematics, chaotic. It is unclear from what Freudenthal writes whether he is implying that Koebe required a wife to help organise his life but certainly he had no wife, remaining a bachelor all his life.
Article by: J J O'Connor and E F Robertson