Mustafa Kemal (Ataturk) emerged from the War of Independence (1919 - 1922) as an national hero, and abolished the caliphate and Shariah laws. (The Thoughtful Guide to Islam, by Shaykh Fadhlalla Haeri)
The one that came right after it and conquered the Iberian peninsula was the Umayyad caliphate. The Umayyad Caliphate was in turn replaced by the Abbassid, Fatimid, Aghlabid, Idrissid, Buyid,...