Japan's benchmark Nikkei index dove more than four per cent by the close of Monday morning trade, Hong Kong was 2.39 per cent lower, Sydney slumped 2.89 per cent and Seoul was off 2.48 per cent.
Fresh fears for the global recovery were stoked at the end of last week after the United States released worse-than-expected jobs figures.
The US announced Friday that 431,000 non-farm jobs were created in May, well below the 500,000 analysts had forecast.
Traders were also fretting about further euro woes after Hungary, which was bailed out in 2008 by the EU and International Monetary Fund, was again facing fiscal trouble.
The markets had reacted after a spokesman for the recently elected Hungarian prime minister hinted that the country could default.
The euro dipped below 1.19 dollars in Asian trade for the first time since March 2006, briefly changing hands at 1.1882 dollars in Tokyo. It also plunged to 108.47 yen, the lowest level in more than eight years.
Credit Agricole strategists said the euro was likely to fall as low as its January 1999 introduction level of 1.1830 dollars later Monday.