Leo Tolstoy's Anna Karenina is not really Konstantin Levin's story, but he does play a prominent role in the lives of many people connected to the title character. He is a man of the land who is always thinking of ways to improve production as well as the relationship between noblemen and peasants. These two things are the core issues of the book he intends to write.
As a landowner in Russia, Levin has always been in a position where he is both responsible for and dependent on the peasants who work on and live on his land. For most of the novel, he is frustrated at the disconnect between his needs and the needs of the peasants. While they do work hard, the peasants work on their own terms and are motivated by things he does not understand. He tries to make improvements on his lands and in the processes of the entire farming cycle, but the peasants remain relatively unmoved by them.
At one point, Levin has a revelation and thinks he understands what might move them: if the peasants were more invested in the process, they would work for themselves as well as him and everyone would be better off for it. He experiments with this theory of cooperative farming and then goes on a tour of Europe to see what countries that are more advanced than Russia have done and are doing int his regard. His book is a compilation both of his research and an explanation of what he feels would improve food production and class relations for the entire country.