Oil was also buffeted, falling to near $70 a barrel, while the dollar fell against the yen and gained versus the euro.
Japan's benchmark Nikkei 225 stock average plunged 349.70 points, or 3.5 percent, to 9,552.69 with investors also cautious before Japan's new leader, Naoto Kan, forms his Cabinet on Tuesday.
South Korea's Kospi lost 1.9 percent to 1,631.85 while Australia's S&P/ASX 200 was down 2.7 percent at 4,329.3. Hong Kong's Hang Seng dropped 2.4 percent to 19,313.05. Benchmarks in mainland China, Singapore and Taiwan also fell sharply.
Selling spread in Asia on fears that Europe's debt problems could spread after Hungarian officials said last week the nation was at risk of a Greek-style fiscal crisis.
The comments shocked investors, pushing the euro to a fresh four-year low against the dollar. Hungary is part of the European Union, but keeps its national currency the forint, which dropped around 5 percent last week.
"The problem seems like a cancerous thing it's spreading from smaller country to smaller country, and many people are afraid that it will spread to a big country like France or Germany, although that's unlikely," said Jackson Wong, vice president at Tanrich Securities, in Hong Kong.
"We don't have major good news on the horizon. We still have crises down the road."
Apart from Hungary's woes, Asian markets were also knocked down by disappointing U.S. jobs data. The U.S. government's May jobs report showed Friday the U.S. economy created 432,000 jobs last month, far fewer than the expected 513,000. Most of the jobs were temporary hiring by the government for the U.S. census.
The result underlined that the U.S. economic recovery is not yet picking up the momentum that investors have been looking for. The Dow Jones industrial average plunged Friday 3.2 percent to 9,931.97. Major indexes all lost more than 3 percent.
Castor Pang, director of research at Cinta International in Hong Kong, said the jobs figures were "much poorer than expected" and a key factor in driving down Asian stocks. Markets were likely to seesaw throughout June.
"Volatility is very great due to lack of confidence," Pang said, calling some investors "very fearful."
In currencies, the dollar fell to 91.34 yen in Tokyo from 91.80 yen in New York late Friday. The euro declined to $1.1925 from 1.1943.
Benchmark crude for July delivery was down $1.14 at $70.39 a barrel in electronic trading on the New York Mercantile Exchange. The contract lost $3.10 to settle at $71.51 on Friday.