In 1879 Moore entered Yale University and took, as he had planned, courses in mathematics and astronomy. His interests became more clearly in the area of mathematics and he received his B.A. in 1883. He remained at Yale to study for his Ph.D. under the supervision of Hubert Anson Newton. Moore's doctoral dissertation was entitled Extensions of Certain Theorems of Clifford and Cayley in the Geometry of n Dimensions and this led to the award of his doctorate in 1885. Newton encouraged Moore to go to Europe for a year and helped to finance the trip. Moore spent the year in Germany, going first to Göttingen where he spent the summer of 1885 studying the German language, but spending most of the academic year 1885-86 attending lectures by Kronecker and Weierstrass at the University of Berlin. Parshall writes in :-
While direct influences of this German study tour on Moore's subsequent mathematical career are difficult to isolate, it is undeniable that Moore returned to the United States with a sense of the importance and desirability of the active and sustained pursuit of original research.It is worth adding that American academics were really forced to train in Europe in Moore's day but when six years later he set up his research school in Chicago it provided for the first time the opportunity for American mathematicians to train in a research-intensive environment in the United States.
On his return to the United States, Moore was appointed as an instructor at Northwestern University for the year 1886-87, then he spent two years as a mathematics tutor at Yale before spending the years 1889 to 1892 back at Northwestern University. During this second spell at Northwestern Moore was approached by William Rainey Harper and offered a post at Chautauqua in New York. Moore refused, knowing that this would mean taking up a post which involved a lot of low level teaching, while he wanted to concentrate on a research. Perhaps it was a fortunate episode since, a year later in 1891, Harper was President-designate of the University of Chicago and wanted to staff the mathematics department with research active staff. Now he could offer Moore the post he wanted to satisfy his research ambitions and Moore quickly accepted.
Moore was appointed professor and acting head of the mathematics department at Chicago when the university first opened in 1892. Prior to this he had married Martha Morris Young on 21 June 1893 in Columbus, Ohio; they had been friends from their childhood days. The Moores had two sons, only one of whom reached adulthood. In 1896 Moore became head of the mathematics department at Chicago, a post he retained until 1931.
When he was appointed at Chicago, Moore persuaded the university authorities to appoint two young German mathematicians Bolza and Maschke to his department. Archibald, in , describes this Chicago mathematics team:-
These three men supplemented one another remarkably. Moore was a fiery enthusiast, brilliant, and keenly interested in the popular mathematical research movements of the day; Bolza, a product of the meticulous German school of analysis led by Weierstrass, was an able, and widely read research scholar; Maschke was more deliberate than the other two, sagacious, brilliant in research, and a most delightful lecturer in geometry. During the period 1892-1908 the University of Chicago was unsurpassed in America as an institution for the study of higher mathematics.
Among Moore's Ph.D. students at Chicago were Dickson, Veblen, Anna Pell Wheeler and G D Birkhoff. Although Robert Moore had Veblen as his supervisor in Chicago he worked with, and was strongly influenced by, Eliakim Moore.
Moore's first main areas of research, which he studied from about 1892 to 1900, were algebra and groups where he proved in 1893 that every finite field is a Galois field. He also studied infinite series of finite simple groups.
In his work on the foundations of geometry begun around 1900 Moore examined the independence of Hilbert's axioms. He reformulated these in terms of points as the only undefined quantities, rather than points, lines and planes as Hilbert had done. His 1902 paper On the projective axioms of geometry showed that Hilbert's axiom system contained redundant axioms.
In  Moore is described as follows:-
Although a gentle man, he sometimes displayed impatience as he strove for excellence in his classes.Parshall in  writes in a similar vein:-
Moore's style of teaching - characterised by a quick-paced presentation of new research ideas and what could be stinging impatience with those who failed to follow - certainly proved effective, at least for the most talented. Moore was a man of high intellectual and academic standards; he expected much from himself, his students, and his colleagues.
The third interest that Moore took up after 1906 was on the foundations of analysis :-
He brought to culmination the study of improper integrals before the appearance of the more effective integration theories of Borel and Lebesgue. He diligently advanced general analysis, which for him meant the development of a theory of classes of functions on a general range. ... Throughout his work in general analysis, Moore stressed fundamentals, as he sought to strengthen the foundations of mathematics.Moore brought precision and rigour to all the fields he studied. Other topics he worked on include algebraic geometry, number theory and integral equations.
In 1893 Moore was one of the main organisers of the first international mathematical congress to be held in the United States. He then approached the New York Mathematical Society with a view to publishing the Proceedings of the congress. In the process he persuaded this society to take a more national role and to change their name to the American Mathematical Society. Setting up a Chicago branch of the Society, which he led, Moore helped in the having the Society reach across the United States. He became a strong supporter of the American Mathematical Society being Vice-President from 1898 to 1900, President from 1901 to 1902 and Colloquium Lecturer in 1906. He acted as an editor of the Transactions of the American Mathematical Society from 1899 to 1907.
In  his contribution is summed up as follows:-
Moore was an extraordinary genius, vivid, imaginative, sympathetic, foremost leader in freeing American mathematicians from dependence on foreign universities, and in building up a vigorous American School, drawing unto itself workers from all parts of the world.Moore received many honours from around the world for his contributions. He was elected to the National Academy of Sciences, the American Academy of Arts and Sciences, and the American Philosophical Society. He received honorary degrees from Göttingen, Yale, Clark, Toronto, Kansas, and Northwestern.
Article by: J J O'Connor and E F Robertson