Definitely yes, and this picture has provided in several Ayah of Quran:
Allah has promised those who have believed among you and done righteous deeds that He will surely grant them succession [to authority] upon the earth just as He granted it to those before them and that He will surely establish for them [therein] their religion which He has preferred for them and that He will surely substitute for them, after their fear, security, [for] they worship Me, not associating anything with Me. But whoever disbelieves after that - then those are the defiantly disobedient. - Al Nur 55
And We have already written in the book [of Psalms] after the [previous] mention that the land [of Paradise] is inherited by My righteous servants. - Al Anbiya 105
But How This Promise of Allah Will Happen?
There are several trustful narrations from prophet Muhammad (PBUH) about this. I provide some from Sunni books, the narrations of Shia books are also very similar:
عَنِ النَّبِيِّ صلى الله عليه وسلم قَالَ " لَوْ لَمْ يَبْقَ مِنَ
الدَّهْرِ إِلاَّ يَوْمٌ لَبَعَثَ اللَّهُ رَجُلاً مِنْ أَهْلِ بَيْتِي
يَمْلأُهَا عَدْلاً كَمَا مُلِئَتْ جَوْرًا " .
The Prophet (PBUH) said: If only one day of this time (world) remained,
Allah would raise up a man from my family who would fill this earth
with justice as it has been filled with oppression. - Sunan Abi Dawud, Book Al Mahdi (37), Hadith 4270
Who Is The Promised Person Who Will Fill Earth With Justice?
We can find the answer to this question also from other narrations of Prophet Muhammad (PBUH):
رَسُولَ اللَّهِ صلى الله عليه وسلم يَقُولُ " الْمَهْدِيُّ مِنْ
عِتْرَتِي مِنْ وَلَدِ فَاطِمَةَ "
The Prophet (PBUH) said: The Mahdi will be of my family, of the descendants of Fatimah. - Sunan Abi Dawud, Book Al Mahdi(37), Hadith 4271
And What Else Will Happen?
There are many other events narrated from Prophet Muhammad (PBUH) and his offsprings about what will happen at that time, including appearance of Dajjal, return of Christ (Isa ibn Maryam) and his joining to Mahdi. Some of them are included in this narration of Prophet Muhammad (PBUH). For longer list of narrations, events, and signs of return of Mahdi you can read the books like one whole chapter of Sunan Abi Dawud called Kitab Al Mahdi - in Arabic and English, المهدي المنتظر (Al Mahdi Al Muntathar) published in Saudi Arabia - in Arabic, or the wikipedia page of Imam Mahdi.