The play is about the last few days in the career of Crocker-Harris, an old classics teacher at a British public school. The man's academic life is fading away and he deeply feels how he has become obsolete. His talk at the end of year prize giving is replaced by the popular sports master and the school will not give him his pension because of his early retirement, although he was depending on it.
When Taplow, a boy who needs Crocker-Harris to pass him so he can go up to the next year, comes to him for help on his Greek, Crocker-Harris begins to show his true feelings.
Crocker-Harris' wife, Millie, is being unfaithful to him with a younger master, something that he has just been ignoring. When Taplow comes to him he admits to himself, and the other man, that he knew about the affair.
As the play ends Crocker-Harris telephones the headmaster saying that he will make his talk at the prize giving, as is his right.
The "Browning Version" referenced in the title is a translation of Agamemnon.
Although the name of the school is not given in the play, it is clearly Harrow School where Rattigan himself was educated, as references are made to a cricket game that is played each year between Eton College and the school.