It is recommended for the Muslim to fast six days of Shawwal, and in this there is great virtue and an immense reward.
“Whoever fasts these six days will have recorded for him a reward as if he had fasted a whole year”, as was reported in a saheeh hadeeth from the Prophet (peace and blessings of Allaah be upon him).
Abu Ayyoob (may Allaah be pleased with him) reported that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Whoever fasts Ramadaan and follows it with six days of Shawwal, it will be as if he fasted for a lifetime.” (Narrated by Muslim, Abu Dawood, al-Tirmidhi, al-Nisaa’i and Ibn Maajah).
The Prophet (peace and blessings of Allaah be upon him) explained this when he said: “Whoever fasts for six days after (Eid) al-Fitr has completed the year: (whoever does a good deed (hasanah) will have ten hasanah like it).”
According to another report: “Allaah has made for each hasanah ten like it, so a month is like fasting ten months, and fasting six days completes the year.” (al-Nisaa’i and Ibn Maajah. See also Saheeh al-Targheeb wa’l-Tarheeb, 1/421).