“A lady and gentleman are making love to one another in the drawing room of a flat in Ashley Gardens in the Victoria district of London.”
The lady is a young widow, Grace Tranfield, in love with the man, Leonard Charteris, who is the ‘philanderer’ of the title. Grace is shocked and disconcerted to find that Charteris, on his own light‐hearted admission, has been in a similar position with Julia Craven and others. The affair with Julia, in fact, has never been broken off. He maintains that it is not his fault that half the women he speaks to fall in love with him; and he is in full flight of cajolery when Julia Craven herself erupts on to the scene, attacks Grace, and announces her intention of staying until Charteris has given her up.
Charteris gets Grace out of the room and unsuccessfully reminds Julia of her supposedly advanced views on marriage. She changes from belligerence to pleading tears, without effect, and, to the consternation of both, the fathers of Grace and Julia enter together. Colonel Craven is suffering from a liver complaint, and much to Charteris’ impatience "has fully made up his mind not to survive next Easter", just to oblige the doctors. Cuthbertson, Grace’s father, is a dramatic critic and theatrically shocked to discover something of the Charteris‐Grace‐Julia triangle; but Charteris explains it is Grace whom he wants to marry.
The scene changes to the Ibsen Club, of which most of the characters are members. A fashionable physician, Dr. Paramore, says he has made a discovery concerning Col. Craven’s fatal complaint, but horrifies Craven’s younger daughter Sylvia by his practice of vivisection. Craven turns up at Cuthbertson’s invitation, and Charteris outrages both men by admitting he had lied to them last night: the truth is, both young women want to marry him, but he does not want to marry either.
Julia enters the Club with Paramore dancing attendance, and manages to trap Charteris alone. She is, however, forced to retire by Sylvia, who delights Charteris by saying Dr. Paramore is in love with Julia. Charteris attempts to flirt with Grace again but is repulsed, and attention is diverted by the distraught Dr. Paramore who has learnt in the British Medical Journal that his ‘discovery,’ Craven’s liver complaint, is a disease which doesn’t exist. He complains of lack of animals for experiment, and resents Craven’s delight at learning he is not to die. Charteris, to cheer him, suggests that Julia is fascinated by him, but it is Grace who comes in first and retires with the doctor privately. Changing tactics, Charteris points them out to Julia, arousing her jealousy. The result is another quarrel between Julia and Grace, who threatens to have Julia expelled from the Club. Julia hurries after Dr. Paramore, to enlist him as a witness in her favor, and Charteris tries to prevent the others following, in order to give the doctor time to propose to her.
This Paramore does, at his house, and Julia, dubious but flattered, accepts him before the others arrive. Charteris is delighted, and Julia and Grace, reconciled, congratulate each other on having escaped him. Nevertheless, Julia bitterly regrets not being brave enough to kill Charteris.
An alternate, and in fact Shaw's original, ending was preserved in manuscript, though not performed until the 1990s. In it the scene at Paramore's house takes place four years later, after his marriage to Julia, when Paramore has tired of Julia and she of marriage. Paramore has fallen in love with Grace and asks Charteris's advice. Eventually, Craven, Cuthbertson and Julia join them and Cuthbertson is persuaded to suggest a solution - divorce, though difficult to obtain in Victorian England without scandal and deceit, is quite easily obtained in South Dakota. Grace joins them, and after a new row between her and Julia, all is agreed. Julia and Charteris are left alone, she presses him to marry her once she is free, but he refuses on the grounds of being a philanderer and no fit husband, and they agree to return to their former ways.

Question: Why do Julia and Grace fight?
Answer: Grace left with the doctor.
Alternative answer: Both are wanting to marry Charteris