Theme in literary terms, refers to a recurrent idea or topic in writing. It is often also the main subject of such writing. In most instances, though, a literary work explores various themes, as does Browning's poem.
The most obvious theme in this monologue is possession. It becomes apparent from the outset that the Duke derives much pleasure from his belongings. He speaks about "my" last Duchess, as if she were an object. He takes pride in talking about the painting he had commissioned. It gives him great joy that the artist, Fra Pandolf, could so accurately depict his previous wife's image on canvas. On a sinister note, though, he comments about the fact that his wife was apparently affected by the artist's complimentary remarks during the sitting.
Was courtesy, she thought, and cause enough
For calling up that spot of joy.
It is apparent that the Duke was offended by the fact that his wife was so easily pleased by the praise or favor of others. He obviously resented her kindness and good nature. He expected these to be reserved for him only. His wife, like a possession, had to be be grateful to him for providing her with such an esteemed title as his. His wife's generosity became too much for him to bear and he had to stop her. The Duke gave instructions and "all smiles stopped together." This implies that he either had her killed or removed to a convent. His action clearly emphasizes his opinion: she was but an object in his control.
The Duke's obsession with owning objects of great value and beauty is further illustrated when he draws his listener's attention to a sculpture of Neptune that he had cast in bronze. The line that most pertinently illustrates the Duke's possessiveness is when he tells his hearer:
his fair daughter’s self, as I avowed
At starting, is my object.
In this instance, the Duke is referring to his future bride - the Count's daughter. The use of the word "object" makes his intention obvious.
Another theme the poem explores is pride. The Duke is evidently impressed by his greatness. His entire speech illustrates an arrogant attitude. He clearly loves being surrounded by things of beauty, for they are a reflection of his self-aggrandizement. When he felt that the previous Duchess could not boost his ego enough, he got rid of her. His vanity is further reflected in his statement that he chose not to stoop in order to beseech his wife to stop what he believed was her inappropriate kindness and care. He felt insulted by the fact that she was not grateful to him for bestowing on her "My gift of a nine-hundred-years-old name." He contends that she treated his esteemed title and rank much as she did any other offering.
On the whole, the poem depicts how pride and arrogance lead the Duke to believe that everything he is surrounded by, even people, are mere possessions.