A degree course at the University of Cambridge is properly called a tripos, after the three-legged stool on which the student traditionally sat to be examined. For most subjects, the degree is taken over three years divided up as Parts I and II, one of which is one year long, the other two years long. Historically, some subjects had a Part III: the degree was optionally taken as the last two years of the standard degree followed by Part III (indeed, one of my contemporaries did just that). Except for the mathematics tripos, Part III had vanished long before my time as an undergraduate (but I gather some faculties have since re-introduced it).
The Mathematics faculty at Cambridge, however, has retained Part III: it serves as a one-year course to prepare new graduates for research (by teaching an enormous amount of material not present in the undergraduate syllabus), ending in an examination in which each student could either fail, pass or achieve distinction. The faculty uses the results (within which they have, but do not release, more refined assessments of candidates) in selecting which students to take on for research.
Several other scientific faculties encourage (in some cases require) their aspiring research students to do Part III of the mathematics tripos as a rigorous final training before starting research. Many research students in related subjects find the courses for Part III sufficiently instructive to warrant attending the lectures even though they are not taking the examinations. From my own experience, I would commend Part III to anyone serious about studying any branch of mathematics in depth: with the warning that it demands hard work.
For my own part, I sat examinations in
having taken courses in these and several other subjects: I was spoilt for choice when it came to looking for interesting things to study. I am not ashamed to have merely passed, nor do I think ill of those who failed: no-one who sat those exams would deny that they were hard !
For a fuller account of the Cambridge degree in mathematics, see the course information provided by the faculty.Written by Eddy.