Hurwitz entered the Realgymnasium Andreanum in Hildesheim in 1868. He was taught mathematics there by Schubert :-
Schubert gave up part of every Sunday to working at geometry with the schoolboy Hurwitz, and the first of the latter's papers, written when he was still at the Andreanum, was a joint paper. It was also Schubert who persuaded Hurwitz's father to allow him to go to university and who sent him with warm recommendations to Klein at Munich.We note that this first paper by Hurwitz, written jointly with Schubert, was on Chasles's theorem. Salomon Hurwitz could not afford to send his son to university but his friend, Mr Edwards, agreed to help out financially, so making a university career for Hurwitz possible. He entered the University of Munich in 1877, before he was eighteen years old, and spent a year there attending lectures by Klein. Although he was greatly influenced by Klein and had already begun to undertake advanced work with him, he went for academic year 1877-78 to continue his studies at the University of Berlin where he attended classes by Kummer, Weierstrass and Kronecker. In particular he attended the one semester course by Weierstrass Introduction to the theory of analytic functions and the notes taken by Hurwitz at this time are reproduced as the book . The lectures contained Weierstrass's version of the arithmetisation of analysis including his "construction" of the real numbers, the ε, δ approach to analysis and his theory of complex functions based on power series.
While at Berlin Hurwitz continued to keep in contact with Klein and assisted him with a paper on elliptic modular functions which he was writing. After three semesters at the University of Berlin, Hurwitz returned to the University of Munich in 1879 to continue working with Klein, so when Klein moved to the University of Leipzig in October 1880, Hurwitz went with him. His Ph. D. was supervised by Klein and he received the degree in 1881 for his dissertation on elliptic modular functions Grundlagen einer independenten Theorie der elliptischen Modulfunktionen und Theorie der Multiplikatorgleichungen erster Stufe Ⓣ.
It would have been natural for Hurwitz to become a Privatdozent at the University of Leipzig since he was a student of Klein, the professor of mathematics there. However there was a difficulty -- Hurwitz did not have sufficient knowledge of Greek to satisfy the Faculty requirements! Luckily Göttingen had no such requirement and Hurwitz became a Privatdozent at the University of Göttingen after submitting his habilitation thesis there in 1882. Hurwitz had not been at Munich during 1881-82, rather he had returned to Berlin where he attended further courses of lectures by Weierstrass and Kronecker.
In 1884 Hurwitz accepted an invitation from Lindemann to become an extraordinary Professor at Königsberg and he was to remain there for eight years. Here he taught Hilbert and Minkowski, becoming a life long friend of Hilbert. Even after Minkowski left the University of Königsberg and went to Bonn, he still returned to Königsberg for every vacation and joined Hurwitz and Hilbert in their almost daily walks :-
During these walks, continued over the whole of the eight year period of Hurwitz's residence in Königsberg, well nigh every corner of the then known mathematical world was explored.In Königsberg Hurwitz met Ida Samuel, the daughter a professor in the faculty of medicine, and they married. The couple had three children: Lisbeth (born 1894),
Eva (born 1896) and Otto (born 1898). Eva began studying mathematics at the ETH in Zürich in 1915, but she dropped out a few years later and turned to extreme-revolutionary' politics, much to her parents dismay. Lisbeth pursued a career as a social welfare worker and Otto studied chemistry.
In 1892 Frobenius left his chair at Eidgenössische Polytechnikum Zürich to return to Berlin and Hurwitz was appointed to the vacant chair at Zürich. Hurwitz remained at Zürich for the rest of his life, unfortunately continually suffering from ill health. His health problems had begun when he contracted typhoid in Munich when he was a student there. The disease was widely spread though the city at that time. In fact he twice contracted typhoid and from then on suffered badly from migraine headaches.
Although, as we have pointed out, Hurwitz remained at Zürich for the rest of his life, that was not because he had not been offered in chair in Germany. Schwarz, who was professor at Göttingen, succeeded Weierstrass by accepting his professorship in Berlin in 1892. Göttingen approached Hurwitz and offered him the vacant chair only weeks after he had accepted the Zürich chair, but he turned down the offer. This must have been a remarkably hard decision for Hurwitz since at that time a chair at a leading German university such as Göttingen would have been much more prestigious to any German than a chair in Switzerland. However Hurwitz was an extremely loyal person, and having given his word that he would accept the Zürich position he would not renege on his promise.
Much of Hurwitz's mathematics can be seen as being strongly influenced by Klein (and also by Riemann whose ideas where transmitted to Hurwitz via Klein). In fact Hurwitz and Klein complemented each other extremely well for the reasons that Young indicates in :-
Klein's strength ... was sometimes regarded as consisting still more in the fertility and the [genius] of his ideas than in the power of developing them.Here then was Hurwitz's strength -- in developing Klein's ideas :-
Klein's new view on modular functions, uniting geometrical aspects such as the fundamental domain with group theory tools such as the congruence subgroups and with topological notions such as the genus of the Riemann surface, was fully exploited by Hurwitz.Hurwitz studied the genus of the Riemann surface. He worked on how to derive class number relations from modular equations. He investigated the automorphic groups of algebraic Riemann surfaces of genus greater than 1, showing that they were finite. He also studied invariant integrals for SO(n, R) and SL(n, R) and Slodowy describes in  how this work, together with Schur's work on orthogonality relations and the character formula for the orthogonal groups, led to Weyl's papers on the representation theory of semisimple Lie groups.
Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations. He also wrote several papers on Fourier series. Soon after he went to Zürich he was asked a question by Aurel Stodola, one of his colleagues, concerning when an nth-degree polynomial with real coefficients
Hurwitz did excellent work in algebraic number theory. For example he published a paper on a factorisation theory for integer quaternions in 1896 and applied it to the problem of representing an integer as the sum of four squares. A full proof of Hurwitz's ideas appears in a booklet published in the year of his death. This involves studying the ring of integer quaternions in which there are 24 units. He shows that one-sided ideals are principal and introduces prime and primary quaternions.
The paper  by Lindström shows another aspect of Hurwitz's work. Here is part of Lindström's summary:-
In 1893 the Swedish actuary and mathematics historian Gustaf Eneström published a theorem on the complex roots of certain polynomials with real coefficients in a paper on pension insurance (in Swedish). This result is now often called the Eneström-Kakeya theorem, since S Kakeya published a similar result in 1912-1913. But Kakeya's theorem contained a mistake, which was corrected by A Hurwitz in 1913. Hurwitz informed E Landau about Kakeya's result (corrected); Landau needed the result in a proof of a theorem on infinite power series. ... We mention a generalization of Eneström's theorem and give an application to a similar result by Hurwitz.The first International Congress of Mathematicians was held in Zürich in 1897.  Hurwitz joined the organising committee at the initial meeting in July 1896 and became one of the main organisers of the congress. He was president of the reception committee, which was responsible for organising the social part of the congress together with the amusement committee, and for providing all congress publications (invitations, programmes, regulations, resolutions, posters, badges etc.) in both German and French.
The reception committee seems to have had the highest workload out of the four sub-committees, and Hurwitz suggested that a publications committee be set up in order to relieve his committee of some of its duties. In the end, the organising committee opted for Rudio's suggestion of enlarging the existing reception committee. Together with Geiser and Minkowski, Hurwitz was responsible for choosing the plenary speakers (on Rudio's suggestion). He also asked Klein to contribute to the preparations. Furthermore, Hurwitz wrote up the attendance list, which was given to every congress participant. He also reviewed the musical entertainment during the steamboat excursion.
On 8th August, Hurwitz spent the entire day 'at the train station, welcoming the arriving mathematicians, handing out the congress cards and organising accommodation for the arrivals if desired'. In the evening, at the collation in the Tonhalle, he gave a welcome speech, in which he stressed the importance of personal relations between mathematicians of different countries.
At the congress itself, he chaired section II: Theory of Functions and Analysis. More importantly, he was one of the four plenary speakers, and the only one of them representing Switzerland. He gave his talk Entwicklung der allgemeinen Theorie der analytischen Funktionen in neuerer Zeit Ⓣ (Development of the General Theory of Analytic Functions in Recent Times), which Hilbert describes as "exemplary due to the clear and concise style as well as the successful selection of this so extensive topic", in the first general meeting on 9th August.
Migraine was not the extent of Hurwitz's health problems which became increasingly severe. His kidneys became diseased and he had one removed in 1905. With only one kidney, and that one not functioning properly, the quality of his life was very poor. Young  writes that:-
... his life [was] one long struggle with a wasting disease. That this struggle was waged with comparative success for so many years appears almost incredible, and can only be accounted for by the constant care and devotion of [his] wife.In  Hilbert's comments on Hurwitz as a person are recorded:-
Hilbert depicted him as a harmonious spirit; a wise philosopher; a modest, unambitious man; a lover of music and amateur pianist; a friendly unassuming man whose vivid eyes revealed his spirit.
Article by: J J O'Connor and E F Robertson