According to the Quran, there are four key obligations on Muslims: two as individuals and two collectively as a Muslim community/state.
1. Fasting/Abstinence (الصیام)
يَا أَيُّهَا الَّذِينَ آمَنُوا كُتِبَ عَلَيْكُمُ الصِّيَامُ كَمَا كُتِبَ عَلَى الَّذِينَ مِن قَبْلِكُمْ لَعَلَّكُمْ تَتَّقُونَ
O Believers! Fasting/Abstinence is prescribed for you as it was prescribed for the people before you so that you may attain TAQWA.
[Al-Quran Surah 2: Verse 183]
2. Making a Will (الوصیۃ)
كُتِبَ عَلَيْكُمْ إِذَا حَضَرَ أَحَدَكُمُ الْمَوْتُ إِن تَرَكَ خَيْرًا الْوَصِيَّةُ لِلْوَالِدَيْنِ وَالْأَقْرَبِينَ بِالْمَعْرُوفِ حَقًّا عَلَى الْمُتَّقِينَ
It is prescribed for you that when death approaches any one of you and he leaves some property behind, then he is to make a will for (his) parents and kindred in a reasonable manner. This is imperative for the true believers
[Al-Quran Surah 2: Verse 180]
3. Fighting (for the cause of Allah) (القتال)
كُتِبَ عَلَيْكُمُ الْقِتَالُ وَهُوَ كُرْهٌ لَّكُمْ وَعَسَىٰ أَن تَكْرَهُوا شَيْئًا وَهُوَ خَيْرٌ لَّكُمْ وَعَسَىٰ أَن تُحِبُّوا شَيْئًا وَهُوَ شَرٌّ لَّكُمْ وَاللَّهُ يَعْلَمُ وَأَنتُمْ لَا تَعْلَمُونَ
Fighting (for the cause of Allah) has been made incumbent upon you although you are averse to it by temperament. It may be that you dislike a thing and (in fact) that is good for you. And it is (also) likely that you like a thing but (in fact) that is bad for you. And Allah knows well but you do not know.
[Al-Quran Surah 2: Verse 216]
4. Equitable retribution in case of manslaughter (القصاص)
يَا أَيُّهَا الَّذِينَ آمَنُوا كُتِبَ عَلَيْكُمُ الْقِصَاصُ فِي الْقَتْلَى الْحُرُّ بِالْحُرِّ وَالْعَبْدُ بِالْعَبْدِ وَالْأُنثَىٰ بِالْأُنثَىٰ
O Believers! Equitable retribution is prescribed for you in the case of murder (or manslaughter): the (same) free man (should be held accountable) if the free man (is the murderer); the (same) slave (should be held accountable) if the slave (is the murderer); the (same) woman (should be held accountable) if the woman (is the murderer)…
[Al-Quran Surah 2: Verse 178]
Note: Although there are several obligations on Muslims, according to the Quran, but this article gives only those obligations for which specifically the Arabic verb “kutiba” (کُتِبَ) is used.