Mr Greenspan, in his capacity as Fed chairman, has primary responsibility for determining interest rates in the US, the world's most powerful economy. As a result, he has the world's stock markets hanging on to his every word.
In recent weeks his words have taken on almost talismanic significance as traders look for clues to help make some sense of the chaos all round them.
The markets surged following a Greenspan speech in which he was interpreted as saying that a co-ordinated cut in world interest rates was on its way. Then they tumbled when the Fed chairman said that wasn't what he'd meant at all.
Investors were cheered when he hinted that, although a co-ordinated rate cut was unlikely, the US could still unilaterally cut rates. But then there was bitter disappointment last week when the 0.25-point cut announced by the Fed was less than the markets had hoped for. This week, the jazz- loving Fed chairman was at it again, hinting that a second cut in US rates was on the cards.
Many traders believe Mr Greenspan is capable both of single-handedly halting the market turmoil, or of plunging the world into a 1930s-style depression. The central banker from Brooklyn is indeed under some pressure.
The septuagenarian Fed chairman, who has made the obfuscatory language of central bankers into something of an art form, started in the private sector running his own economic consultancy.
He moved into politics in the late 1960s as a research director in the Nixon administration, and in 1987 he was appointed chairman of the Fed - just two months before the October stock market crash.
It was a fiery baptism, but Mr Greenspan was, at the time, thought to have handled it well. He moved swiftly to soothe markets and ease monetary policy in the aftermath, and then in 1988 began raising interest rates to stave off the threat of rising inflation.
He is famed for his willingness to strike pre-emptively when he sees inflation risks. "If you wait to see the whites of the eyes of inflation, then it's too late," is one of Mr Greenspan's favourite phrases.
The Fed chairman is often credited with nurturing the long bull run enjoyed by US stocks through much of the 1990s, and his deep knowledge of the US economy impresses many, including Baroness Thatcher. Indeed, Lady Thatcher was said to be so impressed with Mr Greenspan's handle on the economy that, on her return from a trip to the US, she berated Robin Leigh-Pemberton, then Bank of England governor, for what she saw as the paucity of his economic understanding.
But Mr Greenspan's reputation as the doyen of central bankers is starting to look a bit shaky. Revisionists have been less complimentary about his actions in 1987 than the pundits were at the time. Some now even blame the Fed chairman for the subsequent inflationary boom.
There have also been some well-publicised slip-ups along the way, such as his famous "irrational exuberance" speech of December 1996 which failed to dampen booming world stock markets.
Perhaps most importantly, the markets are now starting to question Mr Greenspan's ability to handle the current crisis. His move to cut interest rates last week failed to inspire the markets, and the Fed chairman now finds himself in the position he always sought to avoid - having to react to economic developments rather than make his favoured pre-emptive strikes.
The financial world is waiting anxiously for Mr Greenspan's next step, and investors are hoping he will prove that he has not lost his magic touch.