SAWM: Who is obligated to fast

The scholars are in agreement that fasting is obligatory upon every muslim ‘aaqil (sane) baligh (adolescent) sahih (healthy) muqim (resident), and the woman must be free from haidh (menses) and nifas (post-natal bleeding).
[Fiqh as-Sunah (1/506)]

As for the drop of its obligation with regards to an insane and one who has not reached puberty, the proof is the statement of the prophet (SAW):

قَالَ رَسُولَ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ رُفِعَ الْقَلَمُ عَنْ ثَلَاثَةٍ عَنْ الْمَجْنُونِ الْمَغْلُوبِ عَلَى عَقْلِهِ حَتَّى يَفِيقَ وَعَنْ النَّائِمِ حَتَّى يَسْتَيْقِظَ وَعَنْ الصَّبِيِّ حَتَّى يَحْتَلِمَ

“The pen is lifted from three: the insane until he recovers; the one sleeping until he wakes up; and the child until he reaches puberty.”
[Sahih – Reported by Abu Daud (no.3823)]

And as regards the ill and resident, the proof is the statement of Allah ta’ala:

فَمَنْ كَانَ مِنْكُمْ مَرِيضًا أَوْ عَلَى سَفَرٍ فَعِدَّةٌ مِنْ أَيَّامٍ أُخَرَ

“So whoever amongst you is ill or on a journey (safar), then (he may make up) other days.”
[al-Baqarah: 184]

But if the ill and musafir fast it would suffice them, because the permission to break the fast is only a concession, so if they act according to the original ruling it is better.