Prince Nayef, who was also deputy prime minister and interior minister, had left the country for a holiday and medical tests late last month.
UK PM David Cameron praised the prince's "dedication" and US President Barack Obama focused on his role in fighting terrorism.
He was named crown prince last year succeeding Sultan bin Abdul Aziz.
The succession in Saudi Arabia still passes among the sons of former King Abdul Aziz ibn Saud, who established the modern kingdom during his reign from 1932 to 1953. So far, five brothers have become kings and about 20 are still alive.
King Abdullah, who is 88 years old, had a back operation last year.
Next in line is expected to be Prince Nayef's 76-year-old brother, Prince Salman, who was appointed defence minister in November after spending five decades as governor of Riyadh.
The new crown prince must be approved by the Allegiance Council, a 34-strong assembly of King Abdul Aziz's sons and some of his grandchildren.
Obituary: Prince Nayef
A statement from King Abdullah published by the official SPA news agency, said Prince Nayef had died on Saturday "outside the kingdom" and would be buried on Sunday after Maghreb (sunset) prayers in Mecca.
Unconfirmed reports said Prince Nayef had been receiving medical treatment since May at a hospital in the Swiss city of Geneva.
On 3 June, the deputy interior minister said the prince, who was 77 or 78, was in good health and would return "soon".
Saudi Arabia is expected to declare a period of mourning following Prince Nayef's burial.
The prince was a member of the influential group known in Saudi Arabia as the "Sudairi Seven", which was made up of the sons born to King Abdul Aziz and his favourite wife, Hassa bint Ahmed al-Sudairi.
The kingdom's interior minister since 1975, he led the crackdown on al-Qaeda's offshoot in the country after 11 September 2001.
Personally committed to maintaining Saudi Arabia's conservative traditions based on the Wahhabi doctrine of Islam, he was seen to be more conservative than King Abdullah.
But in 2001, Prince Nayef supported a move to issue women with their own identity cards, a decision which gave women more freedom in many financial and legal transactions, says the BBC's Shaimaa Khalil in Cairo.
He adopted a more stern tone to the Arab Spring, denying claims that it could move to Saudi Arabia, our correspondent adds.
Saudi Arabia's Crown Prince Nayef bin Abdul Aziz Al Saud has died "outside the kingdom", the royal court has said.