The once-a-bandoned abbey of Fiesole, the nestles in the cypress-studded hills outside Florence, is bustling.

Workmen are hanging lamps and tussling with a modern heating system. A new parking lot is packed with cars and motorbikes from most of the countries of Western Europe. The cells once occupied by monks now play host to a third of the 72 graduate students enrolled in a new European university's first academic year.

After 20 years of often ineffectual argument and discussion, the long projected European University Institute, first proposed by Europeanists in the early 1950s, has finally gotten started.

Its goal will be that of "developing the cultural and scientific heritage of Europe in its unity and diversity," or, as some say, building a stronger European consciousness.

Though interdisciplinary seminars and research projects in economics, energy problems, European unification, history, law and political science, the European institute hopes to build up a corps of European expertise.

Major projects - like a North Sea study headed by Prof. Charles Wilson of Cambridge University and a comparative human rights seminar run by Professors Christoph Sasse of the University of Hamburg and Geoffrey Hand of University College, Dublin - are part of an attempt to blend national traditions in a way that may be relevant to tomorrow's Europe.

Marcello Buzzonetti, the institute's secretary general, points out that the idea behind the university's formation was to "bring Europeans together to study and understand Europe's major problems."

Cesar Diaz, a Spanish graduate student interested in political parties, says "we must develop a common European culture since without unification Europe doesn't have much of a future."

Prof. Hand says that once the student body is up to its full annual quota of 300 doctral candidates it will be dispering throughout the continent a substantial group of European-minded individuals who by virtue of the positions they are likely to hold "will have a real impact."

According to Diaz, not all of his fellow-students are convinced federalists. "But just be being here they are on their way," he says. Buzzonetti admits that some of the students are simply attracted by the setting of Florence, some by the university's novelty, some by a desire to finish their doctorates in a hurry.

"But no one is here for a free ride," he says, pointing out that the student scholarships, supplied largely by their national governments, are for the most part inadequate and that living conditions are generally not the best.

Set up the nine Common Market nations with a first-year budget of more than $2 million, the institute has nine well-known European professors on its permanent faculty, a budding library and a large administrative staff to help with all the paperwork caused by having six official languages.

The institute's director, is Max Kohnstamm, a man who has dedicated most of his life to the idea of a unified Europe.

A devoted disciple of European federalist Jean Monnet, Kohnstamm, 62, remembers when Walter Hallstein, then president of the European Coal and Steel Community, first proposed a European university at the 1955 Messina conference that laid the groundwork for the Common Market.

Disagreement among Europe's intellectual elite over whether such a university was really necessary and whether is should be part of Europe's supranational community organizations or, as it now has turned out, the fruit of an intergovernmental accord, stalled things.

"A lot of time was lost in useless bickering," recalls Kohnstamm. In the 1960s the French opposed the concept of a supranational university run by the Common Market. Many European universites were also worried that their own institutes would be reduced to provincial status with the creation of a European academic body.

On the other hand, says Kohnstamm, there has been considerable progress. "Just think," he says, "that since the beginning of the academic year on Oct. 1 students have come to me only to complain about insufficient light, inadequate heating or too few books. No one has said "I'm sitting next to a Frenchman, or a German, or an Englishman, who may have killed my father, - and, after 2,000 years of fighting, that is something quite substantial."