Mr Greenspan was put in charge of America's central bank, the Federal Reserve, by Ronald Reagan in 1987. Since then, the 72-year-old New Yorker has presided over the US economy's longest run of low-inflation growth since the 1950s, and its greatest bull market in shares ever. These triumphs have won him the public's admiration and a reputation for almost god-like economic wisdom. (For some hero-worship, check out the unofficial Greenspan fan club website on http://members.tripod.com/alsfans/index.html.) Such was his popularity that, in 1996, Bill Clinton had no choice but to appoint him for a third term, running to 2000, even though Mr Greenspan is a Republican who used to advise Richard Nixon. The rest of the world's central bankers look to him for leadership - not least in the current financial crisis. And on Wall Street, his words - let alone his changes to American interest rates - can cause share and bond prices to soar or tumble.
Not that the meaning of his words is ever very clear. Occasionally he will coin what in economics passes for a soundbite. In 1996, as share prices rushed ever higher, he warned of the dangers of "irrational exuberance", without actually saying if he thought prices should fall. But usually his speeches are turgid academic essays delivered in a monotone that can produce snores from elderly bankers in the audience. He once joked that on joining the Fed he learnt to "mumble with great incoherence". It is left to financial analysts and journalists to discover what is really on Mr Greenspan's mind by reading between the lines, much as Kremlinologists used to pore over the officialese in Pravda for clues to the inner politics of the Soviet Union.
It is not just his speeches that are studied. Before each month's Federal Reserve meeting on interest rates, CNBC, a cable TV channel that covers business and finance in the style of Sky Sports, monitors the thickness of Mr Greenspan's briefcase as he arrives: slim means no change, fat indicates action. This week, as the dollar plunged against the yen, and stock markets fell almost everywhere, experienced Greenspan-watchers took comfort from his relaxed manner as he lolled against a lectern during a speech. Seriously.
But behind this dour public front is a humorous, down-to-earth libertarian with a taste for the good life. He is well-liked even by his opponents - not least for his patience with those whose grasp of economics is not quite as brilliant as his own. He still plays tennis, with his wife Andrea Mitchell, an NBC-TV correspondent, whom he married only in 1997, after many years together. His first marriage was annulled in 1953.
His free-market philosophy was formed at New York University in the late 1940s, where he became friends with the libertarian thinker Ayn Rand, who believed that government was a barely necessary evil. (Though he is chief regulator of America's financial system, he never misses a chance to say that there is already far too much regulation.) He then set up an economic consultancy, specialising in forecasting for companies. This proved so successful that he ended up on the boards of several clients, including JP Morgan, the blue-blooded investment bank, General Foods and the Mobil oil company. In the late 1960s he discovered a talent for moving in the right Washington DC circles, first advising Nixon during his presidential election campaign, then heading up his Council of Economic Advisers - an influential group of policy wonks - a post he continued to hold under Gerald Ford. In the early 1980s, Ronald Reagan chose him to head an inquiry into the reform of America's state pension system.
Though his time at the Council of Economic Advisers coincided with some of the worst years for the US economy since the Great Depression, remarkably little damage was done to his reputation. It also emerged unscathed from the "Black Monday" stock-market crash of October 1987, only two months after he took charge at the Fed. A few critics blamed him for raising interest rates a month earlier. But he won widespread approval for his swift response to the crash: as the market plunged, he issued a one-sentence statement promising that the Fed stood ready to stump up whatever money was needed to support the financial system. Shares quickly bounced back - in the US and elsewhere - and instead of the global recession that many policy-makers feared would follow the crash, the world enjoyed strong economic growth in 1988. (In Britain's case, thanks also to Nigel Lawson's tax cuts, it was too strong: boom leading to bust in the early 1990s.)
Can Mr Greenspan pull off the same trick a second time to get the world out of its latest current financial-market meltdown? This is his toughest challenge yet. Already the doubters are coming out of hiding: the influential Wall Street Journal recently dared to publish a comment piece headlined, "Alan Greenspan Isn't God".
If he fails now, future historians will blame him for basking in his reputation and doing too little, too late. The arguments will be about what, exactly, he should have done sooner. Some will say that he spent far too much time worrying about the danger of inflation taking off in America, when he should have been cutting interest rates to help Asia's wounded tiger economies. After all, today's financial meltdown has its origins last year in Asia.
Mr Greenspan would probably see things differently. If he made a mistake it was that he did not raise interest rates sharply a couple of years ago, instead of relying on his speeches to talk investors out of their "irrational exuberance". At that time, making money was as easy as falling off a log, even by making what are usually risky investments in dodgy economies such as Russia and Indonesia. People lost sight of the risks, too much money flowed into these risky investments, bubbles formed in their prices, and at the first sign of trouble these have now popped. Maybe if American interest rates had been put up, more money would have stayed at home instead of sloshing around the world, and there would have been no bubbles to burst.
Now investors are running scared, afraid of anything the least bit risky. Mr Greenspan's biggest challenge is to calm them down; his chief difficulty is to say "don't panic" without sounding as nervous as Dad's Army's Corporal Jones. In late September he organised the rescue of Long-Term Capital Management - a fund that had bet billions of borrowed dollars that the prices of a wide variety of assets would move in one direction, only for them to head rapidly the other way. Some critics griped that helping Long- Term's founders, who are old friends of Mr Greenspan, avoid going bust was "crony capitalism". But the main effect of the rescue was to make most investors more frightened than they were before - as for Mr Greenspan to organise such an unprecedented rescue, he must have been terrified about what would happen if the fund had failed. "Substantial damage could have been inflicted on many market participants - [which] could have potentially impaired the economies of many nations," he told Congress. (Kremlinologist's translation: "Oh, !***!!".)
Now things are getting worse. There are signs of more problems at Long- Term Capital. When Mr Greenspan cut American interest rates by 0.25 per cent at the end of last month, investors, who wanted a bigger cut, concluded he was not taking their problems seriously enough. How these problems are handled, as well as which words are inserted between the lines of his next few boring speeches, will show whether Mr Greenspan is really the stuff of legends. Though besieged, he remains our best hope. After he spoke this week to the National Association for Business Economics, he was presented with a tape of Mahalia Jackson singing, "He's got the whole world in his hands".
Matthew Bishop writes for the 'Economist' on finance in New York.