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A weaker yen helped Tokyo's Nikkei add another 0.2% to reach a six-month peak. The index amassed its biggest weekly rise in four years last week.
MSCI's broadest index of Asia-Pacific shares outside Japan added 0.5%, having gained 1.3% on Friday for its best daily rise in almost two months.
Singapore shares gained 0.3% as data showed exports beat forecasts in October.
It is another important week for US monetary policy as Federal Reserve Chairman Ben Bernanke speaks on "Communication and Monetary Policy" on Tuesday.
The day after the central bank releases minutes from its October policy meeting, which will get trawled for hints on when it might start winding back its asset buying program.
Last week, presumptive Fed chief Janet Yellen sounded in no rush to taper, reinforcing market speculation that any move was more likely in March than December.
"If Fed officials think a December tapering is a realistic possibility, some hints to that effect would presumably make their way into the minutes this week," said Michelle Girard, chief US economist at RBS.
"Just making clear that policymakers were open to taking action in December if the economic data showed the impact of the government shutdown was limited would likely suffice to shift expectations."
Girard still believes March is the more likely window for a move, if only because bond markets are typically very thin in December so a taper then could risk major dislocation.
Figures on consumer prices and retail sales are also due on Wednesday and are expected to show that both price pressures and spending were subdued in September.
Speculation over the timing of stimulus tapering has buffeted markets since May when Bernanke first suggested a rollback of the bond-buying programme was not far off.
The Bank of Japan holds its policy meeting on Wednesday and Thursday and is expected to maintain its ultra-loose policy. The BOJ has been perhaps the most aggressive of any major central bank in its asset buying, putting downward pressure on the yen in the process.
The US dollar was trading up at 100.30 yen on Monday, near a two-month high of 100.43. The euro bought 135.20 yen not far from the Oct 22 peak of 135.52, a peak not seen since November 2009.
The single currency drifted off on the US dollar to $1.3478, having edged slowly higher for the past week or so as tapering talk weighed on the dollar. The dollar index edged up to 80.912.
CHINA REFORM
Supporting markets in Asia was cautious optimism about reform in China after the Communist Party unwrapped surprisingly bold reforms late last week.
It pledged to let the market play a "decisive" role in the economy and outlined changes designed to unleash new sources of growth.
China's share index posted its biggest gain in two months on Friday, while the Shanghai Composite climbed 1.7%.
"Multiple growth-friendly measures were announced and represent the biggest freeing up of China's economic policy since the 1990s," said analysts at ANZ in a note.
"Our China economists think that if these reforms are implemented successfully it will substantially reduce the downside risks to China's economy."
In commodity markets, spot gold was steady at $1,287.05 an ounce, having crawled away from last week's trough of $1,260.89.
Brent crude for January delivery eased 25 cents to $108.25 a barrel. US crude for January also shed 25 cents to $94.24, having suffered their sixth weekly drop last week due to a larger-than-expected rise in inventories.
