Here are some of the most commonly cited Quranic verses that directly command or strongly encourage believers not to despair of Allah’s mercy, relief, or hope, even in the hardest times:

1. Surah Az-Zumar (39:53) – The most famous “verse of hope”
   ۞ قُلْ يَـٰعِبَادِىَ الَّذِينَ أَسْرَفُوا۟ عَلَىٰٓ أَنفُسِهِمْ لَا تَقْنَطُوا۟ مِن رَّحْمَةِ ٱللَّهِ ۚ إِنَّ ٱللَّهَ يَغْفِرُ ٱلذُّنُوبَ جَمِيعًا ۚ إِنَّهُۥ هُوَ ٱلْغَفُورُ ٱلرَّحِيمُ
   “Say, ‘O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful.’”
2. Surah Yusuf (12:87) – Prophet Yaqub (Jacob) advising his sons
   يَـٰبَنِىَّ يَغْلِبُوا۟ وَلَا تَيْـَٔسُوا۟ مِن رَّوْحِ ٱللَّهِ ۖ إِنَّهُۥ لَا يَيْـَٔسُ مِن رَّوْحِ ٱللَّهِ إِلَّا ٱلْقَوْمُ ٱلْكَـٰفِرُونَ
   “O my sons, go and find out about Joseph and his brother and despair not of relief from Allah. Indeed, no one despairs of relief from Allah except the disbelieving people.” (The word “rawh” here means relief, ease, spirit of Allah – i.e., never lose hope in Allah’s mercy and help.)
3. Surah Al-Hijr (15:55-56)
   قَالُوا۟ بَشَّرْنَـٰكَ بِٱلْحَقِّ فَلَا تَكُن مِّنَ ٱلْقَـٰنِطِينَ
   قَالَ وَمَن يَقْنَطُ مِن رَّحْمَةِ رَبِّهِۦٓ إِلَّا ٱلضَّآلُّونَ
   They (the angels) said, “We give you good tidings in truth, so do not be of those who despair.”
   He (Abraham) said, “And who despairs of the mercy of his Lord except those who are astray?”
4. Surah Ash-Sharh (94:5-6) – Repeated twice for emphasis
   فَإِنَّ مَعَ ٱلْعُسْرِ يُسْرًا
   إِنَّ مَعَ ٱلْعُسْرِ يُسْرًا
   “For indeed, with hardship [will be] ease.
   Indeed, with hardship [will be] ease.”
5. Surah At-Talaq (65:2-3)
   …وَمَن يَتَّقِ ٱللَّهَ يَجْعَل لَّهُۥ مَخْرَجًا
   وَيَرْزُقْهُ مِنْ حَيْثُ لَا يَحْتَسِبُ ۚ وَمَن يَتَوَكَّلْ عَلَى ٱللَّهِ فَهُوَ حَسْبُهُۥٓ ۚ
   “…And whoever fears Allah – He will make for him a way out
   And will provide for him from where he does not expect. And whoever relies upon Allah – then He is sufficient for him…”
6. Surah Al-Inshirah (94:7-8) (often recited together with 94:5-6)
   فَإِذَا فَرَغْتَ فَٱنصَبْ
   وَإِلَىٰ رَبِّكَ فَٱرْغَبْ
   “So when you have finished [your duties], then stand up [for worship].
   And to your Lord direct [your] longing.”
7. Surah Hud (11:11)
   إِلَّا ٱلَّذِينَ صَبَرُوا۟ وَعَمِلُوا۟ ٱلصَّـٰلِحَـٰتِ أُو۟لَـٰٓئِكَ لَهُم مَّغْفِرَةٌ وَأَجْرٌ كَبِيرٌ
   “Except for those who are patient and do righteous deeds; those will have forgiveness and great reward.”
   (Implies that patience in difficulty is the opposite of despair.)

These verses are the ones scholars and preachers most frequently quote when someone feels hopeless, depressed, or overwhelmed by sin or trials. The strongest and most direct prohibition against despair is 39:53 – many scholars call it “آية الرجاء” (the verse of hope).