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The Hebrew calendar wasn't destroyed, we use it to this day. Today's date is: 14th of Elul, 5771 [September 13, 2011].
The Gregorian Calendar is solar and the Hebrew Calendar is lunisolar.
According to Torah and the Hebrew calendar, it is always on the 10th day of Tishrei, which is the 7th month (Rosh Hashanah is always the 1st day of Tishrei). But because we don't follow the Hebrew calendar, it varies from year to year on our calendar, but is always in early fall.
In the Hebrew calendar, the month of Av is followed by the month of Elul.
Today is Chinese new year
Rosh Chodesh is the first day of the new month on the Hebrew Calendar.
Hanukkah starts on a different day each year because it follows the Hebrew calendar, which is lunar-based, rather than the Gregorian calendar, which is solar-based. Specifically, Hanukkah begins on the 25th of Kislev, the ninth month of the Hebrew calendar, resulting in its date varying between late November and late December in the Gregorian calendar. This shift occurs because the Hebrew calendar is about 11 days shorter than the Gregorian calendar.
Rosh Hashanah is the New Year in the Hebrew calendar and the first of the ten Yamim Noraim or Days of Awe. It falls on the 1st day of the Hebrew month Tishrei, despite the fact that Tishrei is the seventh month of the calendar (the Hebrew calendar is considerably more complicated that the secular calendar), which is late summer/early autumn depending on the year (as the Hebrew calendar and secular calendars do not correspond, a Hebrew date will fall on different secular dates - and vice versa- each year).
The calendar we use today was codified in 360 CE by Rabbi Hillel, but is based on tradition that is at least 2,000 years older than that.
1 week = 1 week on the Hebrew calendar. (The Hebrew week is the same length as the week on the Western calendar)
There isn't any festival celebrated along with Christmas as per the Hebrew calendar, since Christmas is not tied to the Hebrew calendar. (So every year, Christmas falls on a different date on the Hebrew calendar).
The Hebrew calendar is used in Israel, and by Jews all over the world.