Madame Kukshin is a woman in town that Bazarov and Arkady meet through Bazarov's acquaintance, Victor Sitnikov. She is separated from her husband and has made a reputation for herself amongst the young students for being an independent woman and a free thinker. Sitnikov imagines that she and Bazarov will get along well.
When Bazarov and Arkady go to her home, the narrator describes her behavior as follows: "it always struck one that it was the opposite of what she wanted to do; everything with her seemed done on purpose, as children say – in other words, nothing was simple and spontaneous" (13.9).
Bazarov is just as unimpressed as the narrator. He is rude and refuses to engage her in conversation directly. Aside from the airs she has put on, it is clear that Bazarov has no interest in women's rights, a subject in which she is passionately involved. When he goes with her to Matvei Ilyich's ball later on, she is deeply offended that he makes no effort whatsoever to speak with her.
In the survey at the end of the novel, the narrator tells us that Madame Kukshin has settled abroad in Heidelberg, continues to have influence with arrogant young students, and has now moved her interest from science to architecture "in which, so she declares, she has discovered some new laws" (28.11).