Venezuelan President Nicolas Maduro was travelling to offer condolences to Salman, who acceded to the throne on Friday after Abdullah's death at the age of about 90.
US President Barack Obama announced he would cut short a visit to India to travel to the kingdom on Tuesday.
Saudi Arabia has long been a key US ally and last year joined the US-led coalition carrying out air strikes against the Islamic State jihadist group.
From across the Arab and Muslim worlds, from Europe, Asia, and America, presidents, prime ministers and sheikhs have flown in to pay their respects.
It is a recognition of the conservative Islamic kingdom's power as the world's leading oil exporter, a political heavyweight in a region threatened by extremist violence, and as home to Islam's holiest sites.
Salman, a half-brother of Abdullah who reigned for almost a decade, declared today a nationwide holiday "to provide comfort and facilitation to all citizens in offering condolences" and allegiance to the new monarch, the official Saudi Press Agency said.
An AFP reporter saw hundreds of people lined up outside waiting to enter.
They rubbed cheeks and kissed the hands of the goateed Salman, 79, and his heir Crown Prince Moqren, 69.
Ceremonies yesterday moved to the Al-Yamamah Palace, the royal court, where foreign dignitaries greeted Salman and Moqren.
Foreign Minister Mohammad Javad Zarif of Shiite-dominated Iran was among the guests, making a rare visit as Tehran tries to improve relations with its Sunni regional rival.
Both Russian Prime Minister Dmitry Medvedev and Ukraine's President Petro Poroshenko were among the well-wishers, even as pro-Kremlin rebels announced a major new offensive on a strategic government-held Ukrainian port.
Other guests included French President Francois Hollande, Afghan President Ashraf Ghani, Indonesian Vice-President Jusuf Kalla, European royalty and Jordan's King Abdullah II. Prince Charles and Prime Minister David Cameron came from Britain.
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