Yes.
Ali
Hazrat Ali RAU was the last Rightly Guided Caliph. Hazrat Umar bin Abdul Aziz is also considered a Rightly Guided Caliph as he also rules justly.
He is the fourth Caliph Ali Ibn Abou Taleb (Allah be pleased with him)
The last of the 4 Rightly Guided Caliphs was Hazrat Imam Ali. Biography of Hazrat Ali (Sunni perspective) - http://baharemadinah.com/index.php?option=com_content&view=article&id=140:hazrat-ali-bin-abu-talib&catid=47:the-blessed-companions&Itemid=350
Hazrat Ali RAU was the last Rightly Guided Caliph. Hazrat Umar bin Abdul Aziz is also considered a Rightly Guided Caliph as he also rules justly.
The fourth Caliph was Ali ibn Abi Talib (Radi Allahu Anhu).
With the name of almight Allah; leaders of Islam are Muhammad (last prophet of God), his successor Ali (first Caliph/Imam of Islam) and then progony of Ali (Imams).
This depends on what you mean by the word caliph. If you are referring to the rightfully guided Caliphs, who came right after the death of the Prophet Muhammad (S), then the last one, Hasan ibn Ali (RA) died in the year 50 After Hijra. There is debate about whether he or Ali ibn Abi Talib, his father, is the last of the rightfully guided Khalifas. Ali died in 40 AH. The Caliphate, however, was never officially abolished until the year 1924 AD, or 1343 AH. The last Caliph, Abdulmecid the second of the Ottoman Empire, was forced to abolish his title upon the formation of the Republic of Turkey.
Muslims chose Ali in 656.
The question asks about the problems while he is caliph not before being elected as caliph. The first main problem was the opposition of some people to pay the ZAKAT as they were doing in time of the prophet. He waged war against them to oblige them to do as they were doing in time of prophet.Answerwhile he was Caliph Ali (sa) and more then 100 of Sahaba did not accept him as Caliph because they said Allah and prophet have selected Ali (sa) as Caliph at Ghadir. Ali (sa) neve willingly given Bayat to Abubakr. and also many Sahaba.
Caliph Ali was assassinated by Kharijites in 661 C.E., resulting in the end of the Rightly-Guided Caliphs.
There were a few events that he took advantage of to become the leader of the Islamic Caliphate. Firstly, he pressured Caliph 'Ali unreasonably about the assassination of his kinsman, the former Caliph 'Othman. Secondly, after declaring war on 'Ali, he managed to trick 'Ali into declaring that he was unfit to be Caliph which lost him the support of Khawarijites in Egypt. Finally, Mu'awiya defeated 'Ali's armies while (separately) a Khawarijite zealot assassinated him.