Treated to a warm welcome from officials at the kingdom's only airport, the husband and wife team then attended a spectacular reception hosted by the Oxford-educated monarch in their honour at his official residence of Dechenchoeling Palace.
Nestled between India and China in the Himalayas, Bhutan comprises a diverse terrain, ranging from icy mountain peaks in the north to subtropical plains in the south.
One of the world's most isolated nations, its very remoteness has helped ensure the preservation of its vibrant cultural identity.
During their visit, the jetsetting pair explored one of the Buddhist country's 2,000 monasteries, visited an institute for traditional medicine and met with some local youngsters.
Also known as the Land of the Thunder Dragon, the country is a mecca for trekkers and nature lovers, who are drawn by its vast forests and rare flora and fauna.
And the country's capital is equally fascinating.
Planning regulations require all buildings to follow traditional designs, while the city also boasts several colourful festivals, including a popular masked-dance one at the end of summer.