Here are the Gregorian dates for the Biblical Feasts in 2021.
Dates are based on the Jewish (Hillel) calendar.
All observances except Passover begin the prior evening.

Purim
February 26, 2021
(Purim is not one of the Appointed Times in Torah)

This happened on the thirteenth day of Adar and on the fourteenth day they rested, making it a day of feasting and gladness. …The Jews established and took upon themselves, upon their descendants, and upon all who joined with them, that they would commemorate these two days in the way prescribed and at the appointed time every year. These days should be remembered and observed in every generation by every family and in every province and every city.
(Esther 9:17,27-28 TLV)

Passover
March 27, 2021

During the first month, on the fourteenth day of the month in the evening, is Adonai’s Passover.
(Leviticus 23:5 TLV)

Note: The traditional time to begin Passover is in the evening at the end of the day. Some congregations observe it on the prior evening. For a more detailed explanation, go here.

Unleavened Bread
March 28 – April 3, 2021

On the fifteenth day of the same month is the Feast of Matzot to Adonai. For seven days you are to eat matzah. On the first day you are to have a holy convocation and you should do no regular work. Instead you are to present an offering made by fire to Adonai for seven days. On the seventh day is a holy convocation, when you are to do no regular work.
(Leviticus 23:6-8 TLV)

First Fruits
March 29 or April 4, 2021

When you have come into the land which I give to you, and reap its harvest, then you are to bring the omer of the firstfruits of your harvest to the kohen. He is to wave the omer before Adonai, to be accepted for you. On the morrow after the Shabbat, the kohen is to wave it.
(Leviticus 23:10-11 TLV)

Note: The traditional date for First Fruits is the day after the High Sabbath of the Feast of Unleavened Bread, or 16 Aviv. Many congregations observe it on the day after the weekly Sabbath of the Feast of Unleavened Bread, which will change from year to year but always fall on the first day of the week (Sunday). The Counting of the Omer begins on this day. For a more detailed explanation, go here.

Shavuot
May 17 or May 23, 2021

Then you are to count from the morrow after the Shabbat, from the day that you brought the omer of the wave offering, seven complete Shabbatot. Until the morrow after the seventh Shabbat you are to count fifty days, and then present a new grain offering to Adonai.
(Leviticus 23:15-16 TLV)

Note: Shavuot is 50 inclusive days from the day of First Fruits. The traditional date is 6 Sivan. Congregations that observe First Fruits on the day after the weekly Sabbath will conclude the 50-day count on the first day of the week (Sunday) and the date will vary from year to year. For a more detailed explanation, go here.

Yom Teruah
September 7, 2021

In the seventh month, on the first day of the month, you are to have a Shabbat rest, a memorial of blowing (shofarot), a holy convocation. You are to do no regular work, and you are to present an offering made by fire to Adonai.
(Leviticus 23:24-25 TLV)

Yom Kippur
September 16, 2021

The tenth day of this seventh month is Yom Kippur, a holy convocation to you, so you are to afflict yourselves. You are to bring an offering made by fire to Adonai. You are not to do any kind of work on that set day, for it is Yom Kippur, to make atonement for you before Adonai your God.
(Leviticus 23:27-28 TLV)

Sukkot
September 21 – 28, 2021

On the fifteenth day of this seventh month is the Feast of Sukkot, for seven days to Adonai. On the first day there is to be a holy convocation—you are to do no laborious work. For seven days you are to bring an offering by fire to Adonai. The eighth day will be a holy convocation to you, and you are to bring an offering by fire to Adonai. It is a solemn assembly—you should do no laborious work.
(Leviticus 23:34-36 TLV)

Shemini Atzeret
September 29, 2021

On the eighth day there shall be for you an assembly. You are to do no regular work.
(Numbers 29:35 TLV)

Hanukkah
November 29 – December 6, 2021
(Hanukkah is not one of the Appointed Times in Torah)

Then Judas and his brothers and all the assembly of Israel determined that every year at that season the days of the dedication of the altar should be observed with gladness and joy for eight days, beginning with the twenty-fifth day of the month Kislev.
(1 Maccabees 4:59 ESV)

Recommended Resources for Additional Reading

A Christian Guide to the Biblical Feasts by David Wilber
The Feasts of Adonai by Valerie Moody
God’s Appointed Times by Barney Kasden
Messiah in the Feasts of Israel by Sam Nadler
Celebrating Biblical Feasts in Your Home or Church by Martha Zimmerman
Feasts of the Bible Pamphlet by Rose Publishing