His father was an industrialist. Hurewicz attended school in a Russian controlled Poland but with World War I beginning before he had begun secondary school, major changes occurred in Poland. In August 1915 the Russian forces which had held Poland for many years withdrew. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly, a strong school of mathematics grew up in the University of Warsaw, with topology one of the main topics. Although Hurewicz knew intimately the topology that was being studied in Poland he chose to go to Vienna to continue his studies.
He studied under Hans Hahn and Karl Menger in Vienna, receiving a Ph.D. in 1926. Hurewicz was awarded a Rockefeller scholarship which allowed him to spend the year 1927-28 in Amsterdam. He was assistant to Brouwer in Amsterdam from 1928 to 1936. He was given study leave for a year which he decided to spend in the United States. He visited the Institute for Advanced Study in Princeton, New Jersey and then decided to remain in the United States and not return to his position in Amsterdam. Given the impending war in Europe this was clearly a wise decision.
Hurewicz worked first at the University of North Carolina at Chapel Hill but during World War II he contributed to the war effort with research on applied mathematics. In particular, the work he did on servomechanisms at that time was classified because of its military importance. From 1945 until his death he worked at the Massachusetts Institute of Technology.
Hurewicz's early work was on set theory and topology. The Dictionary of Scientific Biography describes it as: "...a remarkable result of this first period  is his topological embedding of separable metric spaces into compact spaces of the same (finite) dimension.*"
In the field of general topology his contributions are centred around dimension theory. He wrote an important text with Henry Wallman, Dimension Theory, published in 1941. A reviewer writes that the book "...is truly a classic. It presents the theory of dimension for separable metric spaces with what seems to be an impossible mixture of depth, clarity, precision, succinctness, and comprehensiveness."
Hurewicz is best remembered for two remarkable contributions to mathematics, his discovery of the higher homotopy groups in 1935-36, and his discovery of exact sequences in 1941. His work led to homological algebra. It was during Hurewicz's time as Brouwer's assistant in Amsterdam that he did the work on the higher homotopy groups; "...the idea was not new, but until Hurewicz nobody had pursued it as it should have been. Investigators did not expect much new information from groups, which were obviously commutative..."
Hurewicz had a second textbook published, but this was not until 1958 after his death. Lectures on ordinary differential equations is an introduction to ordinary differential equations which again reflects the clarity of his thinking and the quality of his writing.
He died during an outing at the International Symposium on Algebraic Topology in Uxmal, Mexico after tripping and falling off the top of a Mayan ziggurat. In the Dictionary of Scientific Biography it is suggested that he was "...a paragon of absentmindedness, a failing that probably led to his death."