On the way into town, you pass the Karl Marx-Hof, among the first workers' flats in Europe, which the right- wing government shelled in 1934. For many, such as the late Richard Crossman, that was the beginning of the Second World War between democracy and fascism.
Then you should stop in the Berggasse, behind the university. It was not only in the arts that Vienna, in the decade before the outbreak of the First World War, was pouring out the ideas by which the 20th century would live - for better and for worse. In the Berggasse, you can visit the apartment in which Sigmund Freud received his patients and wrote the stream of papers and books that created psychoanalysis. It was here that he wrote On the Interpretation of Dreams in 1900; but most of his furniture, his collection of ancient statues and his famous couch are in London, where he had to move in 1938 after Hitler's invasion.
On the other side of the same rather gloomy street of bourgeois apartments is the birthplace of another movement at least as important as psychoanalysis: at No 6 Berggasse is the flat where, in 1898, Theodor Herzl wrote The Jewish State, Zionism's first manifesto. Strangely - or perhaps not - it is not mentioned in any of the guides to Vienna's historic associations.
The city is a museum of the intellectual energy of the quarter-century from 1890 to the beginning of the war, expended particularly by the breakaway artistic movement of the Secession and the Jugenstil (youth style). But, of course, it is much more than that. A royal line in European classical music runs down from Haydn to Mozart, to Beethoven, through Schubert, Schumann, Brahms, Mahler to Schoenberg and the abandonment of classical tonality in the Twenties. Each of these musicians was the pupil of, or at least knew well, his predecessor on the golden ladder, and although not all were born in Vienna, each lived the greater part of his life there.
The heart of musical Vienna lies within two blocks on the Ringstrasse. You cross the street opposite the Opera through a subway, and a block east lies the Musikverein, the great red-brick building that houses the greatest concert hall in the world and a number of other musical institutions, including the Bosendorfer piano company, once one of the the Wittgenstein family's many ventures. (The philosopher, Ludwig, who also designed a house for his sister, which you can visit at Parkgasse in the IIIrd District, near the Landstrasse U-Bahn station, had a brother, Paul, a brilliant pianist. After losing his right arm in the war, Paul taught himself to play with his left hand, and several composers, including Richard Strauss, wrote music for him.) The Musikverein is where all Vienna goes to the New Year's Day concerts of Strauss waltzes.
The Austrians, the joke goes, have persuaded the world that Hitler was a German and Beethoven an Austrian. The fact is that Beethoven lived in Vienna from 1790, when he was 20 - an earlier visit at the age of 17, when he hoped to study with Mozart, was interrupted by his father's death - until he himself died at the age of 56. There are 'Beethoven houses' all over the city. One of the most moving is in the suburban village of Heiligenstadt, where he wrote his famous Testament after learning that he was going deaf. It is next to the church in a pretty square, impossibly crowded by tourists in summer but not too bad in winter; Beethoven was appalled that he could see the church bells, but not hear them.
There are numerous Mozart houses, too. The most interesting is the Figarohaus in a street just behind the cathedral. It contains many mementoes, including the original designs for the costumes of The Magic Flute.
Vienna is a concentric city. At its heart is the Old City, around the cathedral, with its churches and monasteries, courts and alleys, baroque palaces and the 'Bermuda Triangle' of restaurants, bars and nightclubs. Then, on the west side, but still within the ancient walls, is the Habsburg palace, the Hofburg, with more than 2,000 rooms. You can visit the state rooms, the treasury with the imperial jewels, the Spanish riding school and the court churches, as well as the Albertina, which houses the world's greatest collection of drawings. It was in the surrounding streets that The Third Man was filmed; you can visit the locations, sewers and all, on a guided tour.
The inner city is surrounded by the Ringstrasse, shaped like a polygonal horseshoe. Built in the middle of the last century on the glacis outside the old fortifications, it is lined with massive Victorian buildings: the town hall, the Parliament, the university and the great History of Art Museum with the treasures the Habsburgs collected from all over Europe. Within the Ring a second layer of palaces and parks includes the Belvedere palace of Marlborough's brother-in-arms, Prince Eugene, with its priceless collection of Klimts, Schieles and Kokoschkas. Then come the districts, the Bezirke, each with its own character; Josefstadt, for example, the VIIIth District, was the Jewish one. And beyond lies a layer of garden suburbs, such as the Hohe Warte in the north or Hietzing to the west, built by the cultivated upper middle class at the turn of the century.
Finally, in a great fan around the north, west and south sides of the city is the Wienerwald, the Vienna Woods. Vineyards rise gently to the wooded crest of the hills, and from the higher points, such as the Leopoldsberg and the Kahlenberg, you can enjoy the view back over the city.
Four Vienna experiences should not be missed. The city has two opera houses: the Staatsoper, or State Opera, the successor to the Court Opera, is the well-known one; and the Volksoper, the 'People's Opera', the equivalent of the Coliseum in London. (Vienna, with 1.6 million people, has about 50 theatres: about the same number as London, which has four times as many people.)
The second, less spiritual but none the less ecstatic, experience is the konditorei, the confectionery. The two most famous are Sacher's, behind the Opera, and Demel, in the Kohlmarkt. They are at daggers drawn over which sells the authentic Sachertorte, a chocolate cake with apricot jam.
Third, there are the heurigen. The word means 'this year's wine', and vineyards in the suburbs sell their product on the premises. Some, especially in Grinzing, have become boring stop-offs for tourist buses; others are pretentious and expensive. But if you can find a genuine heurigen off the beaten track - most Viennese have a secret one they will take you to if they like you - it is an unforgettable experience. The wine is white, from last year's grapes, and you can order food - slices of roast pork, cheese or olives. They bring jugs of wine - straight or diluted with fizzy water as g'spritzte, origin of the American 'spritzer' - to your long, scrubbed wooden table. In some heurigen, they sing, but perhaps it's better just to talk and let the wine mellow your mood.
Best of all, though, are the Viennese coffee-houses. No two are alike. They have different architecture, decor, coffee, cakes, sandwiches and, above all, clientele. In Hawelka, the atmosphere is intellectual; in Landtmann, opposite the Parliament and close to the Burgtheater, the clientele is smarter, and business people hold formal meetings at the tables.
Our favourite is the Cafe Museum, designed by Adolf Loos, but a bit shabby now, in the Karlsplatz. It is full of young people, writing their university essays, reading, talking and spinning out their cup of coffee, with or without a star-shaped dollop of whipped cream. This is life with the cream: mit schlag]
Getting there: British Airways and Austrian Airways offer direct flights to Vienna.
Accommodation: Hotels are expensive since the Black Wednesday devaluation (the Austrian schilling is pegged to the German mark). The two classic palace hotels of the Habsburg era are the Imperial, patronised by the Habsburgs themselves, across the street from the Musikverein, and the Bristol, beloved of the old nobility, across from the Opera. There are lots of more modest hotels, however. We stayed in the Karntnerhof, a comfortable and friendly hotel in the medieval heart of the Old City, for pounds 56 a night (double room with bath ).
Food: Solid, not to say heavy, but well-cooked. Two excellent places frequented by younger people in the inner city are Oswald und Kalb, in Backerstrasse, and the Salzamt, in the 'Bermuda Triangle'.
Opera: Opera tickets run from pounds 8.80 for standing, leaning on a rail, to pounds 88 in the middle of the dress circle; tickets at pounds 15 offer a perfectly acceptable view: do not dress: you will see everything from full fig to jeans.
Getting around: For under pounds 7, a three-day ticket entitles you to travel on the (excellent) U-Bahn (underground), streetcars and buses.
Other attractions: Some of the 'musts', other than those mentioned in the article, are the museums (ethnography, musical instruments) in the Hofburg palace complex; the Kunsthistorisches Museum, on the Ring; the Hundertwasserhaus, a crazy masterpiece of kitsch built by the painter and environmentalist Friedensreich Hundertwasser in the Lowengasse in the IIIrd District; the Karl Marx-Hof, the pioneering public housing project, 1,000 yards long, that the government shelled to break the resistance of the workers in 1934; and, 20 minutes away by train, the great monastery at Kloster Neuburg.
The best collection of Klimt, Kokoschka and Schiele in Vienna is at the Upper Belvedere, though the Museum of the History of the City of Vienna in the Karlsplatz is worth a visit. Do not miss two small but perfect buildings in the Karlsplatz: Olbrich's Secession gallery, with Klimt's magical frieze; and Otto Wagner's white-and- gold Karlsplatz station. A gem in the suburbs is the tiny station (Hofpavillon) that Wagner built for the emperor near the summer palace at Schonbrunn. Three cemeteries are worth a visit: the great Zentral Friedhof with its deserted Jewish section, in the XIth District; the St Marxer cemetery, in the IIIrd District, of the Biedermeier period (early 19th century), where Mozart was reburied.
The great baroque palaces are mostly around the Herrengasse, and the most splendid of many baroque churches is the Karlskirche, in the Karlsplatz. Do not miss the auditorium of the Old University, a superb rococo baroque building, now the Academy of Sciences, or the nearby Jesuit church.
Apart from the coffee-houses mentioned, try the Cafe Brunerhof, in the Stallburgstrasse, smoky and melancholic; the Cafe Mozart, centre for spying in the Third Man era; and Cafe Tuchlaub, in the Tuchlaubstrasse in the Old City.
Information and books: The Vienna Tourist Board provides useful information, including Vienna A to Z, a 'guide to sightseeing on your own'. Architecture in Vienna published by Prachner, is an invaluable guide to 350 buildings, Gothic, baroque, 'Ringstrasse style' Jugendstil and modern in the city and suburbs. But the real find is the series of books by Christian Nebehay, with neat potted biographies of the greats of Vienna, with notes on where they lived and worked. One, Vienna 1900 Architecture and Painting, covers Otto Wagner, Klimt, Kokoschka, Schiele, Olbrich, Hoffmann, Moser and Loos; one is on the classical composers; and a third is on Music around 1900, covering Berg, Wolf, Mahler, Bruckner, Brahms, Schoenberg and others.