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Currently there is no present caliph of Islam as Caliphate was ended in 1923 after dissolution of the Ottoman Empire, a consequence of WWI but some Shia sects have present Caliph (Khalifa). First caliphate of Islam called Rashidun Caliphate after the last prophet of Islam, Muhammad ﷺ (Peace Be Upon Him).
Abu Bakr (Allah Please with him) was the first Caliph of Islam whereas the last caliph Abdülmecid II of Ottoman Caliphate.
Present Caliph in Shia Muslims
Shia Muslims consider Ali (Allah Pleased with him) as the first Caliph or Imam. Shia who believe in 12 Caliphs or Imams to arrive after the last prophet Muhammad ﷺ and the 12th imam is the present imam (or present caliph) who disappeared and will reappear when Jesus (Peace Be Upon Him) will arrive back to the world from heavens.
Ismaili Muslim Caliph
Ismaili Muslims know to be Present Imam sect of Islam. Prince Karim Aga Khan is the current present Imam of Ismaili Muslims only. For understanding purpose only, Ismaili Caliph or Imam is supposed to be just like Roman Pope of Catholic Christians only and rest of the Christians sects e.g. Orthodox, Protestant etc. do not accept Roman Pope as their religious leader.
Bohri Muslims (Dawoodi Bohra) also have faith on present Caliph or present Imam but on At-Tayyib Abu'l-Qasim as the present Caliph or Imam.
Mirza Masroor Ahmad and Ahmadiyya Jamat Caliph
Mirza Masroor Ahmad is the Ahmadiyya Caliph and nothing to do with Caliphs or Caliphate of Islam ever. Muslims are agreed and declare followers of Ahmadiyya Jamat as non-Muslim group and also Islamic scholars of all sects have consensus that Ahmadiyya, Ahmadi or Qadiani/Qadiyani are non-Muslims.
On the internet, Ahmadi Khalifa (caliph) is associated with Islam which is wrong and misleading information.
Final Caliph or Imam Mahdi
Caliph or Imam Mahdi (to be arrived) are known to be last Caliph of Islam in Sunni Muslims whereas Shia Muslims consider Hujjat Allah ibn al-Hasan as Mahdi and the last and the present Caliph of Islam to reappear at the time of the rerrun of Jesus (Peace Be Upon Him).
Replied Article in Islam