The Prophet (PBUH and HF) said: "Even if the entire duration of the world's existence has already been exhausted and only one day is left (before the day of judgment), Allah will expand that day to such a length of time, as to accommodate the kingdom of a person from my Ahlul-Bayt who will be called by my name. He will fill out the earth with peace and justice as it will have been full of injustice and tyranny (by then)."
References:
Sahih al-Tirmidhi, v2, p86, v9, pp 74-75
Sunan Abu Dawud, v2, p7
Musnad Ahmad Ibn Hanbal, v1, pp 84,376; V3, p63
The Prophet (PBUH and HF) said: "al-Mahdi is one of us, the members of the household (Ahlul-Bayt)."
Reference: Sunan Ibn Majah, v2, Tradition #4085
The Prophet (PBUH and HF) said: The Mahdi will be of my family, of the descendants of Fatimah (the daughter of the Prophet (PBUH)).
References:
Sunan Abu Dawud, English version, Ch. 36, Tradition #4271 (narrated by Umm Salama, the wife of the Prophet)
Sunan Ibn Majah, v2, Tradition #4086
The Prophet (PBUH and HF) said: "We (I and my family) are members of a household that Allah (SWT) has chosen for them the life of the Hereafter over the life of this world; and the members of my household (Ahlul-Bayt) shall suffer a great affliction and they shall be forcefully expelled from their homes after my death; then there will come people from the East carrying black flags, and they will ask for some good to be given to them, but they shall be refused service; as such, they will wage war and emerge victorious, and will be offered that which they desired in the first place, but they will refuse to accept it till they pass it to a man from my family (Ahlul-Bayt) appears to fill the Earth with justice as it has been filled with corruption. So whoever reaches that (time) ought to come to them even if crawling on the ice/snow since among them is the Vice-regent of Allah (Khalifatullah) al-Mahdi."
References:
Sunan Ibn Majah, v2, Tradition #4082,
The History Tabari
al-Sawa'iq al-Muhriqah, by Ibn Hajar, Ch. 11, section 1, pp 250-251
Abu Nadra reported: We were with the company of Jabir Ibn Abdillah... Jabir Ibn Abdillah kept quite for a while and then reported Allah's Messenger (may peace be upon him) having said: "There would be a Caliph in the last (period) of my Ummah who would freely give handfuls of wealth to the people without counting it." I said to Abu Nadra and Abu al-Ala: Do you mean Umar Ibn Abd al-Aziz? They said: NO, (he would be Imam Mahdi).
References:
Sahih Muslim, English version, v4, chapter MCCV, p1508, Tradition #6961
Sahih Muslim, Arabic version, Kitab al-Fitan, v4, p2234, Tradition #67
One who takes lessons from the events of life, gets vision, one who acquires vision becomes wise and one who attains wisdom achieves knowledge.