For Muslims, the Islamic New Year begins on the first day of Muharram, the first month in the Islamic calendar. Last year that was on the 18th December 2009.
The first year was the year during which the Hijra, i.e. the emigration of prophet Muhammad (pbuh) from Mecca to Medina, occurred. This was a very significant event for the Muslims since it was on that day Muslims began to organise their society in Medina upon Islamic law (shariah) for the very first time.
Dates for the next few years of Islamic New Year:
# 1431 AH: 18 December 2009
# 1432 AH: 7 December 2010
# 1433 AH: 26 November 2011
# 1434 AH: 15 November 2012
# 1435 AH: 4 November 2013