Donald Trump was hailed with an applause and cheers as he and President Moon greeted each other in the South Korean capital.
Trump handshakes have made the headlines before with his 19-second long handshake with Japanese Prime Minister Shinzō Abe.
The US leader’s trip is designed to support the military alliance against North Korea and strength in unity is the message they want to send to North Korean leader Kim Jong-un, according to our correspondent.
The US President said the threat of North Korea will be at the “front and centre” of discussions with South Korea during his five-nation Asia tour.
Mass-protests have taken place in South Korea’s capital against Donald Trump’s military threat against North Korea.
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North Korea’s nuclear goals were on the agenda during the US President’s visit to Seoul.
During a joint press conference on Tuesday, Mr Trump sent a direct warning to Kim Jong-un, telling him he is prepared to “defend ourselves and our allies”.
Speaking directly to South Korea, he said: “We are partners and friends who have fought side by side in a war and worked hard towards great lasting peace.
“We will not allow North Korea to threaten all that we have built, and all that we have built together.”
Raising fears of the outbreak of all-out war, he said North Korea was “a threat to people of South Korea and people all across the world”.
He said: “We will together confront North Korea. We will be prepared to use the full range of our military capabilities if needs be.”
South Korea’s spy agency said last week that North Korea may be preparing another missile test, raising speculation that such a launch could be timed for President Trump‘s trip to the region.