The next step is a postgraduate degree, and for those seriously intent on original research, a masters or doctorate by research, rather than a taught course, is preferred. Roughly a quarter of postgraduate students gain their qualification by research alone, and the total numbers have begun to rise again after a fall-back in the early Nineties. In 1994-95 there were 87,000 research postgraduates; last year, 94,000.
So how do this year's brightest young graduates set about finding a place from which they can launch their glittering careers? With the greatest difficulty, according to Professor David Warner, pro-vice-chancellor of the University of Central England (UCE). There is no Ucas for postgraduates and Professor Warner says that most of them find the system quite impenetrable.
Three years ago UCE tried to set up a database to link aspirant researchers with the 30,000 or more academics in more than a hundred institutions who are able to supervise research projects. There were very few takers.
"I think the need is ever greater," Prof Warner says, "The `market' for postgraduate places is growing inexorably and it is more and more important to make it simpler for the most brilliant students to find the right place to start their research careers."
At the moment, he says, most students "cop out" by staying in the institutions where they have done their first degree. But this assumes that the university offers research opportunities in a relevant field, and it leaves out of account the brilliant overseas students who have traditionally come to the UK to further their own learning - and, incidentally, Britain's reputation as a centre of academic excellence.
So how do students who want or need to move institution find their niche? Some newspapers publish lists of research opportunities, and some institutions advertise their vacancies. But on the whole they offer only a subject or a topic for which they have Research Council studentships available, and do not, as Prof Warner thinks they should, say anything about the supervisors who are free. Only a minority of advertisements provide full details of a research project and information on who will supervise it.
With the market tightening as employment opportunities for graduates improve, Prof Warner thinks that universities must urgently improve their recruitment strategies. For overseas students in particular, competition is worldwide, and anything that makes it harder for postgraduates to find appropriate places is damaging for British institutions and ultimately for Britain herself.
The University of Manchester Institute of Science and Technology (UMIST), which is one of the most heavily research-based institutions, was so concerned about the post-graduate lottery that it commissioned a survey of its students to discover what had influenced their choice of institution.
This showed that there is competition for postgraduates places, with half the students having applied for places elsewhere. But there is variation between subjects. There was more competition for places in computing, electrical engineering and languages than there was for chemistry and management subjects.
Surprisingly, perhaps, the survey indicated that publicity - in the form of brochures, posters and advertisements - had very little effect in bringing postgraduates to UMIST. Successful applicants were much more likely to have been influenced by personal contacts with UMIST staff or former students, or lectures and talks by staff.
The reasons for choosing UMIST were diverse. Some wanted to be in trendy Manchester. A small number had applied purely "on spec". But most had applied because they knew that UMIST had a good reputation for their subject. For others, it was the only option that satisfied their particular interest and for some, especially in management and biochemistry, it was an opportunity to work with a particular professor.
UMIST's survey covered postgraduates doing research and taught courses, and it was noticeable that those looking for a research-based place were the least satisfied with the way they had found it. A clear majority found that brochures failed to answer all their questions, and more than half had to seek further information about fees, finance, accommodation and their research topic.
Prof Warner remains convinced that there must be a better way. Three factors, he thinks, will make the situation worse over the next few years, the first of them simply the inexorable increase in the numbers of young people gaining first degrees. If the same proportion want to gain postgraduate qualifications, then postgraduate places must increase to keep pace or competition will become more intense.
At the same time, he says, a first degree as a qualification for some jobs has been devalued by the rising number of graduates. This is not the same as saying that standards have fallen, simply that employers looking for some mark of distinction among applicants will increasingly look for a postgraduate qualification.
But, conversely, Prof Warner says, the fact that the academic profession is becoming more and more international and universities are opening up all over the world means that competition from other countries for PhD students and for staff is growing. Overseas students and staff will not come to Britain unless British systems are made user-friendly.
"We have the technology to improve our systems, just as we have the means to teach and research in innovative ways, using new technology and part- time options," Prof Warner concludes. He thinks that we just need to get on with it