Karl Menger attended the Doblinger Gymnasium in Vienna (1913-1920) where one of his fellow students was Pauli. Karl entered the University of Vienna in 1920 to study physics. However Hahn became a lecturer in Vienna in 1921 and Menger attended a course he gave on What's new concerning the concept of a curve.
Menger became interested in the topic and was encouraged by Hahn to work on the topic. Menger's work led him to a definition of dimension independently of Urysohn, however Urysohn had died in a drowning accident before he could publish his work and Menger was not aware of it.
After a severe lung disease which forced Menger to spend more than a year in a sanatorium, he returned with important papers he had written on dimension while in the sanatorium and completed his doctorate in 1924.
In 1925 Menger was invited by Brouwer to take up a post in the University of Amsterdam where he spent two years working with Brouwer. In 1927 Menger was invited by Hahn to accept the chair of geometry at the University of Vienna when Reidemeister left for Königsberg. Menger was not sorry to leave Amsterdam since he had become involved in a priority dispute with Brouwer and they were not on the best of terms. In 1938, as a result of the political situation in Austria, he resigned his chair and accepted a post in the USA at the University of Notre Dame. Karl Signund writes:-
After the war, the reconstruction of the bombed-out State Opera was accorded highest priority by democratic new Austria. Men like ... Menger, however, were politely told that the University of Vienna had no place for them.
At Notre Dame Menger organised a Mathematical Colloquium based on that at Vienna. He arranged a visit by Gödel to Notre Dame but failed to persuade him to accept a post there. However after the war began to affect the USA in 1941, academic life was disrupted and Menger's Mathematical Colloquium failed to become influential as the Vienna Circle had been.
Around this time Menger's interests in mathematics broadened and he began to work on hyperbolic geometry, probabilistic geometry and the algebra of functions. Menger's work on geometry failed to have the impact that his work on dimension theory had. This is almost certainly because geometry, at this time, was a rather neglected area of mathematics, particularly in the USA.
In 1948 Menger went to the Illinois Institute of Technology and he was to remain in Chicago for the rest of his life.
In  his interests are described as follows:-
He had a great love of music. ... he built up a notable collection of decorative tiles from all over the world. ... he ate meat sparingly, particularly in his last years. But he was always glad to sample cuisines, from Cuban to Ethiopian, that were new to him. He liked baked apples.
Article by: J J O'Connor and E F Robertson