Well...technically there have been many self-proclaimed Caliphs in the present day, although none are at all widely accepted by the greater Muslim world. The ruler of the Ottoman Empire was nominally the the caliph, however he was deposed when his Empire fell at the closing of WWI. As we know, the Ottoman Empire's successor state is the modern nation of the Republic of Turkey, and its first leader, Mustafa Kemal (Ataturk) constitutionally abolished the post of the caliph as one of the founding acts of the republic. There was a summit later about a decade later to discuss a potential revival of the caliphate, however, interest at the time was low, (as most Muslim nations at the time were involved in secular state-building or had secular independence movements), so the effort did not amount to much.

 

sources  

Kazancigil, Ali and Ergun Özbudun. Ataturk: Founder of a Modern State  

Kedourie, Elie. "The End of the Ottoman Empire", Journal of Contemporary History

In additions to what others have said, Hussein bin Ali, the King of Hejaz following WWI, proclaimed himself Caliph of all Muslims as his kingdom included the Holy Cities of Mecca and Medina. Additionally, as he was a Hashemite, he claimed direct descent from Muhammad.

However, following the defeat of Hejaz to the Saudi Sultanate of Nejd, no one made a substantive claim to the caliphate.