For a few days in August, 1969, on a dairy farm in upstate New York, a half-million young people got together to hang out, dance, and listen to music. Forty years later, for many people, the Woodstock Music & Art Fair is the defining event of the Sixties. For LIFE photographer John Dominis, covering the festival became one of the most moving adventures of an amazing 25-year career. "I was much more interested in the people who were there than the musicians," he tells LIFE.com. "I liked the music okay, but I really liked the kids, and what they were doing, and how they felt about it all. I just wanted to capture that." Here, Dominis' favorite photographs from Woodstock.