Japan's benchmark Nikkei 225 eased 0.7% to 20,972.71 while South Korea's Kospi rose 0.4% to 2,098.71. Hong Kong's Hang Seng picked up 0.9% to 27,481.62. The Shanghai Composite index was 0.1% lower at 2,884.46. Australia's S&P ASX 200 added 0.6% to 6,570.00. Shares rose in Taiwan and in most of Southeast Asia.
The U.S. Federal Open Market Committee will hold a two-day meeting starting Tuesday. Traders will keep a close watch on a policy statement, to be released Wednesday, and a news conference held by Fed Chairman Jerome Powell shortly after that.
Powell will likely stick to the theme of a speech he made earlier this month: That the Fed will act if it thinks the Trump administration's trade conflicts are threatening the U.S. economy.
Financial markets had rallied on the remarks, which fueled hopes that the Fed would cut interest rates this year. Although a rate cut isn't expected this time around, fresh comments from Powell could support another wave of buying or have investors swing the other way.
"With scant data releases and no significant developments on the trade front, the mood in Asia will likely remain cautious with investors awaiting the FOMC meeting and possible adjustments to the dot plots before taking on substantial bets," ING economists Nicholas Mapa and Prakash Sakpal said in a commentary.
Central banks in Britain and Japan will also announce their latest monetary policy decisions this week.
On Tuesday, the Reserve Bank of Australia released minutes from a policy meeting in June, which saw it easing its cash rate to 1.25%. According to the minutes, members agreed that further rate cuts were "more likely than not" in the period ahead, although there were other ways to reduce unemployment.
Trading was choppy on Wall Street on Monday, weighed by uncertainty over the U.S.-China trade dispute.
But indexes finished the day with slight gains. The S&P 500 index edged 0.1% higher to 2,889.67 and the Dow Jones Industrial Average also rose 0.1% to 26,112.53. The Nasdaq composite climbed 0.6% to 7,845.02. The Russell 2000 index of smaller company stocks added 0.7% to 1,532.75.
ENERGY: Benchmark U.S. crude dropped 17 cents to $51.76 a barrel. The contract gave up 58 cents to $51.93 a barrel on Monday. Brent crude, the international standard, shed 20 cents to $60.74 a barrel.
CURRENCIES: The dollar slipped to 108.27 Japanese yen from 108.53 yen. The euro strengthened to $1.1234 from $1.1220.