The Prophet s said, “Whoever fasts Ramadan and follows it with six days from Shawwal it is as if they fasted the entire year.” (Sahih Muslim)
Based on the above hadith the majority of scholars, hanafi, Shafi`i, and Hanbali schools, state that it is recommended to fast six days of the month of Shawwal.
It is permitted to fast these six days separately throughout the month, or consecutively, after Eid al-Fitr and they consider it prohibitively disliked (makruh tahriman – hanafis) and sinful to fast on Eid day itself.