Born in New York City, New York to Jewish immigrants from Russia, Papp founded the New York Shakespeare Festival in 1954 with the aim of making Shakespeare's works accessible to the public.

In 1957 he was granted the use of Central Park for free productions of Shakespeare's plays at what was named "The Delacorte Theater". By age 41 after the Delacorte had become a solid institution, Joe looked for an all-year theater he could make his own. After looking at other locations, Joe fell in love with Lafayette Street's Astor library for its location and character. With massive renovations in order, Joe moved his staff to his newly named Public Theater hoping to attract a newer, less conventional audience to new and innovated playwrights.

Papp also obtained the use of the Astor Library Building in 1967; this has since become known as the Joseph Papp Public Theater.

Joe depended on authors more than anyone and saw them in a much higher respect then actors or even directors, and his focus moved farther away from the Shakespearean classics. With plays such as Charles Gordone's No Place to Be Somebody (the first African American dramatist and off-Broadway show to win the Pulitzer Prize) and the plays of David Rabe, Tom Babe, and Jason Miller, Joe brought the Public into a new phase. “…With the new playwrights, the whole direction of the theater changed. Joe changed direction and none of us realized for a while that he had changed direction. The Public Theater became more important than the Delacorte. The new playwrights became more interesting to Joe than Shakespeare. “ (Ming Cho Lee, Festival Designer)

Papp is known for his productions of Hair, The Pirates of Penzance, and A Chorus Line.