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Whenever I have a chance," says Zygmunt Bauman, "I wholeheartedly advise my friends to retire from active university service as soon as possible. As for my enemies, I would recommend them to stay in it as long as possible ... Landing at the other end of long university service enables one to do more, much more of all those things which brought one into it in the first place. Learning and teaching go on as before, but now unimpeded, uninterfered and un-tinkered with."

There are people who say goodbye (or something ruder) to their colleagues at their retirement party, take to the golf course and never think about work again. Others turn up bright and early the next morning. Many academics go on working productively and happily long after the rest of their age group have been put out to pasture.

Indeed, some find it the most productive stage of their career. Bauman, emeritus professor of sociology at the University of Leeds, is a particularly inspiring example.

At 82, he is widely regarded as one of the world's leading sociologists and a "prophet of postmodernism". He has averaged well over a book a year since retiring in 1990 and has recently focused on the "liquid", provisional identities common in a world where even cosmetic surgeons offer loyalty points, on the assumption that we may wish to reinvent ourselves several times in the course of a lifetime.

Retirement has allowed Bauman to do far more of the things he has always wanted to do, "since, for the privilege of doing them inside the university walls, one had to pay a rather heavy price of endless and exhausting, yet barren and eminently forgettable, brawls and squabbles in and out of committee rooms, hours (days, months ...) spent on form-filling and on composition of reports, censorship from the top obsessed with 'efficiency' and jealousies of promotion-obsessed peers".

Bauman believes that those who work in the humanities and social sciences are "in an enviable position". He says: "We need only libraries, computers and a few bright minds around as the tools of creative work to test our findings." In the experimental sciences, "the present of knowledge devalues its past, and burns bridges behind at accelerating speed". The humanities, in contrast, are "engaged in the process of perpetual recapitulation of ideas that hardly ever exhaust their cognitive powers, turn truly obsolete and are doomed to be confined to oblivion".

And that means, Bauman concludes, that "the volume of knowledge possessed by scholars working in the humanities tends to grow with their owners'/carriers' age. The size of accumulated experience and the length of memory are assets that may well outweigh the common liabilities of old age."

Retirement can also give academics the opportunity to reinvent themselves - without resorting to the surgeon's knife. Distinguished psychologist Peter Herriot, who has had professorships at City University and Birkbeck, University of London, did extensive research and consultancy in occupational areas such as selection and the employment relationship, although he has also studied language development, Down's syndrome and memory.

Yet there were always limits, he says, to how far he could range while employed. "It is next to impossible to make a radical change in your area of expertise in the middle of your career. You have to stop publishing in your original area for three years at least while you get a good enough conceptual grasp of the new area to understand what research questions need to be asked," he says.

It was only with retirement that Herriot was able to do the intensive reading that allowed him to contribute to an important field that drew on his "whole life experience" as well as on his professional expertise.

He must be unusual among academic psychologists in having been brought up in a (Christian) fundamentalist family. "So when fundamentalism became a world issue after 9/11," he explains, "I realised that I might have something to offer, since you have to use all the psychology you know plus a whole lot more to understand fundamentalists. You have to look at their social context as well as their thought processes to get a broad enough picture."

Herriot has now published an academic study on the subject, Religious Fundamentalism and Social Identity (2007), while a book for a more general audience, Religious Fundamentalism: Global, Local and Personal, is due to be published this year.

Peter Warr, emeritus professor at the University of Sheffield's Institute of Work Psychology, has also "gained from the opportunity in retirement to reflect and make connections between issues". Some of his own research has focused on wellbeing and job satisfaction among older people and the differences between "early retired and late- employed individuals". His decision to continue working has not only added to the body of knowledge but helped him to carry out one of the common "tasks" of ageing.

"Having time and inclination, I've been able to try to bring together a lot of my earlier projects, finding an overall 'whole'," Warr says. "Reviewing one's previous activities and 'making sense' of them is usual at older ages. I recently published a substantial review of the whole wide field of Work, Happiness, and Unhappiness, creating an integrated framework that is different from any other. It may not be completely correct, but I'm pleased to have been able to make progress in the right direction."

Academics who want to continue teaching or conducting research after they get to retirement age obviously have to make pragmatic arrangements with individual universities.

Tony King, professor of British government at the University of Essex, for example, has twice renewed his contract for three years since he reached 67 and still has what amounts to a full-time role, lecturing to first-year students, holding weekly two-hour study groups and carrying out research.

He has recently published a major overview of the British constitution and continues to work on a book about US politics. He is also involved, with his former vice-chancellor Ivor Crewe, in the study of a topic that grows larger almost daily, policy failures in British politics.

Jack Spence, another political scientist, believes: "You become a better teacher with age." He is now in his 51st year as an academic and says that "health and strength permitting, I'll go on thinking, reading and writing". He worked as a professor of politics and later pro vice-chancellor at the University of Leicester (1973-91) and then as director of studies at Chatham House (1991-97).

In his mid-sixties he was offered a one-year contract as a visiting professor in the department of war studies at King's College London, teaching a popular course on the theory and practice of diplomacy since the end of the Cold War. The contract has been renewed each year for a decade. Spence has an additional role as academic adviser at the Royal College of Defence Studies while continuing to research a major book on the end of apartheid in his native South Africa.

Mary Evans, emeritus professor of women's studies at the University of Kent, retired in November after 37 years. But she went straight into another role - a three-year contract as a visiting fellow at the London School of Economics' Gender Institute, where she is researching books on detective fiction and the 21st-century vogue for "misery writing".

Spence, Evans and King have clearly negotiated one-off mutually beneficial contracts with their employers. Universities may well experience increasing requests for such arrangements.

Alison Johns, human resources director at the London School of Economics, foresees a rise in retirement numbers. She tries to give sympathetic consideration to those who wish to retain their connections with the school or to continue working after 65. "At the LSE people continue to have access to libraries, the senior common room and dining room. Some professors and readers become emeritus and can then get shared office space. Others formally retire but are then re-engaged on a part-time basis, especially in specialist areas."

Recent legislation has added an extra dimension to this issue. The Employment Equality (Age) Regulations (2006) gives staff the right to ask to continue working beyond their normal retirement age. Although employers are not obliged to comply with such requests, they will need to be seen to consider them and to make a business case for rejecting them.

But not all academics who want to continue researching, writing and sometimes teaching have had such happy experiences. Geoffrey Cantor, emeritus professor of the history of science at the University of Leeds, argues: "Through their publications and participation at conferences, retired academics often bring considerable kudos to their universities and feature in research assessment exercise submissions." Yet many, he says, "can be left out of pocket and sometimes exploited".

"Since retirement," he explains, "I have joined the army of retired academics in the arts and humanities who, out of commitment to their subjects, have remained research-active. They bring much expertise to their fields of research. But I have been disappointed at how little financial support there is, either through my own university or through the main funding bodies.

"Although Leeds has generously allowed me to retain my e-mail address and access to electronic resources, I have paid for all of my other research expenses. Currently I am assessing whether I should attend two conferences in Oxford over the summer, which together will cost nearly ?700. At best, my ex-department may help with a small proportion of the cost."

Even more striking is the experience of David Knight, emeritus professor of the history and philosophy of science at Durham University. When he retired in 2002 his position was not filled. Concerned that his specialist subject, the history of science, could disappear from the syllabus altogether he stepped in and saved the day by offering unpaid teaching for the next two years.

By agreeing to do this, Knight assured continuity and kept the topic "alive" until the department was expanded and a successor eventually appointed in 2004. He still does some voluntary teaching, but that is a purely personal choice rather than a decision he felt obliged to take out of loyalty to his subject.

Knight doesn't seem embittered by any of this. But such cases do raise financial and indeed moral issues. If an academic works in a post for 30 years and then continues for another ten, pensioned but otherwise unpaid, it can clearly make a difference to the prestige of a department or institution as well as the body of research knowledge. Yet some universities seem to make little effort to encourage them to keep working.

Someone in a position similar to Knight's, but who feels a lot less happy, is Richard Gombrich, Boden professor of Sanskrit at the University of Oxford from 1976 to 2004. He is now governor of the Oxford Centre for Hindu Studies and has a paid role as general editor of the Clay Sanskrit Library.

"In order that Buddhist studies should not disappear from Oxford University," he explains, "I founded the Oxford Centre for Buddhist Studies. I am the academic director and a former pupil of mine, Geoff Bamford, retired from a business career, is executive director. Not only do we work for nothing, but so far almost the entire expenditure for our infrastructure has come out of our own pockets. The university would never give us a pencil or allow us to make a free local telephone call.

"But things are far worse than that. For instance, it was made a condition of our recognition as a 'recognised independent centre' that we take out insurance against malfeasance by our employees - even though we have no employees! Every step has been like wading through treacle.

"Like the Hindu Centre, we arrange lectures, classes and tutorials for the university. Tutorials are paid for by the colleges; lectures and classes are mostly, but not invariably, provided at no cost to the university, and we in effect pay the teachers. My own teaching, however, I have been providing gratis.

"How do we pay teachers? We are engaged full-time in fundraising. In my last years as a professor fundraising was one of my main duties and occupations; since my retirement it has only got worse. Were it not for voluntary help from outside (not only from retired people) the university would not be able to provide certain subjects. Nor is one exactly overwhelmed by expressions of gratitude ..."

This is a sad story, but it is matched by many others of luckier academics still able to pursue their passions.

"What drives me? Curiosity, I suppose," says Bauman. "The same urge that sent me 60 years ago to study philosophy and sociology. Curiosity is a curious urge; one that is set to grow more insatiable as one strives to feed it. Ageing is of little help here. Once married, life and curiosity can't be separated ..."