Japan's Nikkei 225 stock average rose 1.1 percent to 15,551.28 and Hong Kong's Hang Seng added 0.2 percent to 23,732.43. Seoul's Kopsi climbed 0.8 percent to 2,021.52 and Australia's S&P/ASX 200 gained 0.5 percent to 5,360.20.
Going against the trend was Thailand where the SET index fell 0.7 percent after anti-government protests on the weekend drew tens of thousands of people. China's Shanghai Composite rose 0.3 percent to 2,203.47.
Appetite for stocks was energized by Wall Street's string of milestones and an easing of political risk signaled by the first significant progress in years to curtail Iran's nuclear ambitions.
Iran on Sunday reached an agreement with the U.S., Britain, France, Russia, China and Germany to limit enrichment of uranium to 5 percent, far below the level needed for nuclear weapons. Iran got limited relief from sanctions that have hobbled its economy, but an embargo on its oil exports remains in place while negotiations continue for an enduring deal.
"Perhaps the Iran nuclear deal, effectively setting limits to Iran's nuclear program, has added to the buoyant risk mood," said Stan Shamu, market strategist at IG in Melbourne, Australia.
Wall Street continued to be lifted by the Federal Reserve's super easy monetary policy, signs of gradual improvement in the U.S. economy and rising company profits.
The Standard & Poor's 500 closed above 1,800 for the first time Friday, capping seven straight weeks of gains. The Dow also logged a seventh week of increase.
The S&P 500 rose 8.91 points, or 0.5 percent, to 1,804.76. The Dow Jones industrial average gained 54.78 points, or 0.3 percent, to 16,064.77 after Thursday passing 16,000 for the first time. The Nasdaq composite jumped 22.49 points, or 0.6 percent, to 3,991.65.
In energy trading, benchmark U.S. crude for January delivery was down 83 cents at $94.01 in electronic trading on the New York Mercantile Exchange as the nuclear deal made it more likely that the sanctions choking Iranian oil exports will eventually be lifted. The contract fell 60 cents to close $94.84 on Friday.
The euro was barely changed at $1.3543. The dollar rose to 101.75 yen from 101.29 yen late Friday.
AP From Bangkok