British Prime Minister David Cameron, Prince Charles and French President Francois Hollande were among the first leaders expected while US Vice-President Joe Biden was due to visit in the coming days.
Abdullah, a cautious reformer who led the US-allied Gulf state through a turbulent decade in a region shaken by the Arab Spring uprisings and Islamic extremism, died early yesterday aged about 90.
Since he took the throne in 2005, Riyadh has been a key Arab ally of Washington, last year joining the coalition carrying out air strikes against the Islamic State jihadist group.
Gulf rulers and leaders including Turkish President Recep Tayyip Erdogan and Pakistani Prime Minister Nawaz Sharif were among those who attended Abdullah's simple funeral at Riyadh's Imam Turki bin Abdullah mosque today.
Also Read
The late king's body, wrapped in a cream-coloured shroud, was borne on a simple litter by members of the royal family wearing traditional red-and-white checked headgear.
The body was quickly moved to nearby Al-Od public cemetery and buried in an unmarked grave, in keeping with tradition.
In the evening hundreds of Saudis queued to enter a royal palace where they rubbed cheeks and kissed the hands of their new leaders, in a symbolic pledge of allegiance.
Officials did not disclose the cause of Abdullah's death, but the long-ailing ruler had been hospitalised in December with pneumonia.
President Barack Obama was quick to pay tribute to Abdullah yesterday as a "valued" ally, and the State Department said Washington saw no indication that cooperation would change.
Biden said on Twitter he would lead a delegation to Saudi Arabia "to pay respect and offer condolences".
In his first public statement as king, Salman, 79, vowed to "remain, with God's strength, attached to the straight path that this state has walked since its establishment".