Commenting http://blogs.telegraph.co.uk/news/tomchiversscience/100278157/a-jug-of-wine-a-loaf-of-bread-and-thou-is-this-the-islamic-caliphate-isis-imagine/:

When the 20th century began, there were four political leaders calling themselves Caliph. Ottoman Caliph, surviving until 1924; the Zaidi Caliph, surviving until 1962; the Sokoto Caliph, surviving until today but mostly using the title of sultan after the fall of the Sokoto Caliphate in 1903, and the Caliph of the break-away Mahmudiya sect, surviving until today.

That was no historical aberration - a thousand years ago there was a Caliph in Baghdad, Cairo and Cordoba.

Caliph is "just" a title and the actual powers and ideology of a Caliph can be widely different. As a result modern Caliphate could be anything from ISIS' jihadist dream to British Commonwealth -alike or an Islamic EU or UN.

When Omar Khayyam lived, the Caliph in Baghdad was just a ceremonial ruler with little power, secular or religious. The real power in the Middle East was in the hands of the Seljuk sultan. The Caliphs, just like the Japanese emperors, were more often than not just figureheads.