ʿĀshūrā’ is the 10th day in the month of Muḥarram, which is the first month in the Islamic lunar calendar. This year it will be on Thursday Aug 19, 2021. Based on several authentic statements of the Prophet Muhammad, it is strongly recommended you fast on this day.
ʿAbdullah ibn ʿAbbās related that he had never seen the Prophet so keen on fasting any other day, outside of Ramaḍān, than on the day of ʿĀshūrā’. The Messenger of Allah emphasized fasting on this day to such an extent that he once said, “I hope that Allah would forgive the sins of the previous year for the one who fasts on the day of ʿĀshūrā’.”
Near the end of the Prophet’s life, he wanted Muslims to differentiate themselves when he heard that Jews and Christians also commemorated it. So he said: “Next year, God-willing, we will fast on the 9th [as well]”, but he passed away the following year. After a detailed analysis of all the teachings of the Prophet relating to fasting on the day of ʿĀshūrā’, Imām Ibn Ḥajar al-ʿAsqalānī [d. 852 A.H./1449 C.E.] concluded that there are three different ways to fast this day, in order of ascending virtue: 
- Fast only on the 10th day
- Fast on both the 9th and 10th day [fasting the 10th and 11th is also fine though he didn’t mention that]
- Fast on the 9th, 10th, and 11th
One of the misconceptions that has occurred in the minds of many Muslims is that the 10th of Muḥarram should be observed as a day of mourning because Ḥusayn ibn ʿAlī, the righteous grandson of the Prophet, was killed on that day. It is true that the day he was killed was indeed sad and a great tragedy. However, many righteous people have been killed on other days, yet none of those dates have been taken as days of mourning.
Ḥusayn’s own father, ʿAlī, was also killed unjustly in the month of Ramadan in the year 40 A.H. but no one has specified that day as a day of mourning to be observed by Muslims. Furthermore, the Messenger of Allah witnessed the slaying of many of his family members who were very dear to him such as his uncle Ḥamza, his adopted son Zayd ibn Ḥārithah, and his cousin Jaʿfar [ʿAlī’s brother], yet he never appointed a day of mourning for them. Therefore, although it may be beneficial to reflect on the martyrdom of Ḥusayn and the historical lessons that can be derived from such an incident, there is no basis for singling out his death as a day of mourning.
 Ṣaḥīḥ al-Bukhārī 3:43, #2006
 Ṣaḥīḥ Muslim 2:818, #1162; Muslim scholars have explained that this refers to minor sins only.
 Another Ḥadith explains: “The Prophet entered Madīnah and observed the Jews fasting on the Day of `Āshūrā’. He asked, ‘What is this?’ They said, ‘This is a righteous day. It is the day that God saved the Children of Israel from their adversary, so Moses fasted.’ The Prophet said, ‘We have more right to Moses than them.’ So he fasted and ordained fasting on that day.” Muslim scholars have often speculated what fast the Jews were observing on the 10th of Muḥarram circa 622CE as recorded in the Ḥadith. Was it Pesach (Passover), Yom Kippur, or a lesser fast like Asher b’Tevet? However, the dates seem to not coincide between Passover and `Āshūrā’ until over a decade after the Hijrah and even then Passover falls on the 18th of Muḥarram, and not the 10th. Shaykh Abul Ḥasan ʿAlī Nadwī clarifies the confusion as follows: “Besides, any attempt to make the Lunar Arabian Calendar correspond to the Solar Jewish Calendar can, at the best, be only hypothetical. The ancient custom of intercalation has also taken a hand in adding to the confusion. It is incorrect to question the authenticity of successive Ḥadīths merely on the basis of an erratic and inconstant calendar. It is also possible that the Jews of Madīnah were different from the other Jewish communities where the fast of ʿĀshūrā’ was concerned and observed it with greater enthusiasm and regularity, and, in this respect, they were similar to the Arabs who, seeing that so many important events had taken place on that day, fasted on it out of reverence. ʿĀishah said, “The Quraysh fasted on the day of ʿĀshūrā’ during the Age of Ignorance and the Prophet also kept it.” [Ṣaḥīḥ Muslim] Further, the fast days among the Jews living in different countries differed from one another. We have seen how in the Jewish Encyclopedia it is indicated that apart from the fixed fast-days many fasts of a local or national character had become established among the Jews from the early days, which varied from place to place. Private fasts were also common among the Jews and one could take it upon oneself to fast on certain days in memory of certain events or at the time of adversity to arouse God’s mercy. In these circumstances, it is quite possible that the fast of ʿĀshūrā’, on the tenth day of the first month of the Arab Calendar, was peculiar to the Jews living in Arabia alone. Perhaps, it is for this reason that the Talmud and the Jewish Calendar are silent on this score. But their judgment is influenced by the ignorance of the habits and practices of the Jews living in various parts of the world, especially in Arabia where they had been settled for generations as a distinct community, possessing their own beliefs and customs and receiving local impressions in the historical course of things.”
 Ṣaḥīḥ Muslim 4:798 #1134a
 Ibn Ḥajar al-ʿAsqalānī, Fatḥ al-Bārī Sharḥ Ṣaḥīḥ al-Bukhārī, 4:245