Who Is Required to Fast?

Like salah, the fasts of Ramadan are also farz for every mature and sane Muslim man and woman.
It is stated in Dur-e-Mukhtar that fasts became obligatory on 10th Sha’ban two years after hijra. (Dur-e-Mukhtaar ma’a Rad-dul-Muhtaar, V3, P330)

The Reason of Fasts' becoming Obligatory

Most of the acts in Islam remind us of faith-strengthening Islamic and historical events. Running between Safa and Marwa reminds us of Sayyidatuna Hajira's Radiallaho Ta’ala ‘Anha; walking and running between these two mountains 7 times in search for water for her son Sayyiduna Ismaeel Alaihi salam. Allah azwajal liked this sacrifice and made it necessary (wajib) for all those who perform Hajj and Umrah to imitate her in order to keep this act alive. Likewise, Our Beloved Rasul Sallalahu Alaihi Wasallam spent some days of Ramadan in seclusion in the cave of Hira where he Sallalahu Alaihi Wasallam used to refrain from eating during the day and worship Allah azwajal at night. In order to bring back the memory of this blessed act of His Beloved Rasul Sallalahu Alaihi Wasallam Allah azwajal made it obligatory for us to fast in Ramadan.

The Fasts of the Prophets Alaihi salam

The people of the previous ummahs also fasted but their modes of fasting were different from ours.

According to narrations, Sayyiduna Adam Alaihi salam would fast on the 13th, 14th and 15th dates. (Kanzul-Umaal, V8, P257, Hadith 24188)

Sayyiduna Nooh Alaihi salam used to fast every day. (Ibn-e-Maja, V2, P333, Hadith 1714)

Sayyiduna Isa Alaihi salam fasted every day and would never miss a single fast. (Kanzul-Umaal, V8, P304, Hadith 24624)

Sayyiduna Dawood Alaihi salam used to fast on alternate days. (Sahih Muslim, P584, Hadith 1189)

Sayyiduna Sulaiman Alaihi salam used to fast on the first three, the middle three and the last three days of every month (9 days in total). (Kanzul-Umaal, V8, P304, Hadith 24624)