The modern attempt at reconstructing one of the ancient Attic Calendars has the New Year beginning with the month of Hekatombaion, and "with the first sighting of the new moon after the summer solstice. Ideally, the solstice would occur toward the end of Skirophorion, the final month of the year." That New Moon will occur on July 17th this year. The year has its start when the first sliver of the New Moon becomes visible.
The Calendar at Hellenion.org has the 1st of Hekatombaion taking place on Wednesday the 19th, though Tuesday the 18th after nightfall will mark the change. This coming year will be the 3rd year of the 700th Olympiad.