With Francois Truffaut (1932-1984), Godard was among the best known of the New Wave filmmakers.
Godard made numerous short films during the 50s but Breathless is his first full length film. Breathless was deeply influenced by American Film Noir (particularly Touch of Evil) but breaks from American continuity style (look for jump cuts and breaking of the eyeline match rule).
Next: Breathless
Godard was strongly influenced by existentialist philosophy and Marxism. He was a child of the middle class who became interested in film and film criticism at university (unlike Truffaut who was essentially a child of the streets). Godard wanted to create a new realism in film. Additionally, he was deeply influenced by playwright Bertol Brecht's idea that art and human problems are solvable when they are viewed rationally.
Throughout the 1960s, Godard continued to make generally well received films, some of which are considered classics including Contempt (1964) and Pierrot le Fou (1965).
After 1968, Godard became increasingly political, first Marxist, then Maoist. He was convinced that "one cannot think" with traditional narrative films. So, his become more fragmented and disjointed. They became calls to political action and contemplation of the relationships between symbols, stories, and reality (and difficult to watch). Today, filmmakers often think very highly of these films but most others don't watch them.