Munk studied at the University of Copenhagen, where he began his first produced play, En idealist (1928; Eng. trans. Herod the King). This was originally misunderstood by the critics, although it was acclaimed 10 years later.

In 1931 his Cant (on the rise and fall of Anne Boleyn) was a success, and Ordet (1932; The Word), a miracle play set among Jutland peasants, established him as Denmark's leading dramatist. Ordet later was made into a motion picture by the Danish director Carl Dryer. For his principal character, Munk often chose a dictator, or “strong man,” whom he showed struggling in vain against God. En idealist, Ordet, and Han sidder ved smeltediglen (1938; He Sits at the Melting-Pot), a drama of Adolf Hitler's Germany, are his three best plays, though several others, such as Cant, Kærlighed (1926), and I Brændingen (1929), remain popular.

Five Plays, with preface and translations, was published in 1953. Munk was a conscientious and much-loved parish priest, and during World War II his outspoken sermons drew large numbers of persons to the ranks of the Resistance and led to his being killed by the Nazis.