Monetary easing by the European Central Bank last week has brightened the mood for risky assets globally, with an upbeat US nonfarm payrolls report released Friday giving further impetus.
On Wall Street overnight the S&P 500 ended at a fourth straight record closing high and the Dow at its third.
Tokyo shares gained 0.1%. MSCI's broadest index of Asia-Pacific shares outside Japan was up 0.2% after touching 492.11, its highest since July 2011.
"The week ahead is likely to be a quiet one for emerging market asset prices as the focus shifts to the World Cup," strategists at Brown Brothers Harriman wrote in a note to clients.
"Still, central banks continue to occupy the centre stage after Mexico's surprise cut and Brazil's extension of its forex intervention program last week. Many are looking for the Korean central bank to step up its forex interventions as USD/KRW breaks below the key 1020 level," they said.
The South Korean won strengthened to below 1020 to hit a near six-year peak against the dollar on Monday, supported by persistent stock inflows and in catch-up trade after the European Central Bank eased policy, although participants were wary of official intervention.
The dollar continued to benefit from rising US Treasury yields. The dollar index, which measures the greenback's strength against a basket of key currencies, was little changed after rising 0.2% on Monday.
The dollar stood little changed at 102.51 yen. The euro was flat at $1.3589 after shedding nearly 0.4% on Monday.
In a week without hard-hitting data, the market was keeping an eye on Chinese inflation data due at 0130 GMT. Consensus is for China's CPI to rise 2.4% year-on-year in May, a level economists reckon would not be high enough to change the central bank's stance on monetary policy.