By Agence France-Presse
Trump said he had formed a "special bond" with the North Korean leader, who reaffirmed his commitment to "complete denuclearisation of Korean Peninsula" in a joint agreement signed by the two.
While the meeting was not expected to see any immediate results, it has provided hope for peace on the Korean peninsula.
However, investors remain concerned about a possible global trade war after the weekend's Group of Seven summit in Canada ended with Trump withdrawing support for a joint communique and accusing host Prime Minister Justin Trudeau of being dishonest.
That came just after he had hit Canada, Mexico and the European Union with steel and aluminium tariffs, sparking threats of retaliation that some fear could escalate.
For most of the day eyes were on Singapore, where Trump and Kim became the first sitting leaders of their countries to meet.
The pair signed what Trump called a "pretty comprehensive" and "very important" document, which spoke of "new US-DPRK relations" and committed Washington to "security guarantees".
Kim said the two Cold War foes had vowed to "leave the past behind" and promised "the world will see a major change".
'Air of optimism'
Stocks fluctuated through the day but by the end of trade Tokyo was up 0.3 percent and Shanghai closed 0.9 percent higher. Hong Kong was 0.4 percent higher.
However, Seoul dipped 0.1 percent, while the Korean won eased 0.2 percent.
Singapore was slightly lower, as were Wellington, Kuala Lumpur and Taipei.
In early trade London and Paris each rose 0.3 percent, while Frankfurt gained 0.8 percent.
"After quickly shrugging off the G7 fight club in Quebec, investors latched on to the air of optimism circulating from the Singapore summit," said Stephen Innes, head of Asia-Pacific trading at OANDA.
"And with pro-euro signals emanating from the newly formed Italian government placating investors anxiety about the future of the eurozone, the global risk was in the happy zone Monday."
However, Goohoon Kwon, co-head of Korea research and senior Asia economist at Goldman Sachs, told Bloomberg Television ahead of the signing that markets had already factored in a lower risk of conflict on the Korean peninsula.
He added that the "follow through, execution, implementation" of any agreements was most crucial.
While the Trump-Kim summit was the big news story, traders are keenly awaiting the Fed and ECB policy meetings. The US bank is expected to lift interest rates Wednesday but its post-meeting statement will be closely watched for a clue about its plans for future hikes.
Then on Thursday, European bank officials will likely debate for the first time cutting back on their crisis-era stimulus programme.
The euro rallied Monday on the back of comments from new Italian Finance Minister Giovanni Tria ruling out Rome's exit from the single currency. That eased concerns about his populist, Eurosceptic government's intentions regarding the currency bloc.
However, the single currency dropped slightly in Asia, while the pound was also under pressure as British MPs prepare to vote on a string of amendments to key Brexit legislation that could force Prime Minister Theresa May's hand in talks with the European Union.
Key figures around 0720 GMT
Tokyo - Nikkei 225: UP 0.3 percent at 22,878.35 (close)
Hong Kong - Hang Seng: UP 0.4 percent at 31,196.94
Shanghai - Composite: UP 0.9 percent at 3,079.80 (close)
London - FTSE 100: UP 0.3 percent at 7,760.24
Euro/dollar: DOWN at $1.1763 from $1.1785 at 2100 GMT
Pound/dollar: DOWN at $1.3349 from $1.3378
Dollar/yen: UP at 110.34 yen from 110.02 yen
Oil - West Texas Intermediate: UP 37 cents at $66.47 per barrel
Oil - Brent Crude: UP 31 cents at $76.77 per barrel
New York - Dow Jones: FLAT at 25,322.31 (close)