Like all good literature which stands the test of time, Crime and Punishment is a reflection of human nature and it is true today as it was when it was written because the essential nature of man does not change. Raskolnikov commits an awful crime as a kind of experiment, something which is something most of us do; however, we do all, at times, commit senseless and hurtful acts we wish we could take back or undo. As Raskolnikov tries to assuage his guilt, he grows sick and delirious; most people with a cense of conscience are somehow impacted by their bad behavior, as well. There was a wrestling for his soul, a battle Raskolnikov finally won--but it was a close thing. Raskolnikov's experience after the murders is, in the broadest sense, a common experience for all of us. That makes what happened to him relevant to all of us.
The primary theme of "Crime and Punishment" is guilt. Rudya kills the old woman, and spends the rest of the novel sick with guilt and fear. Another theme of the novel is the connection between the mental and physical self. Although our protagonist's pain is psychological, he experiences severe physical illness as well. The readers must ask what the significance of this connection is. Are our physical and mental selves separate or the same? To what extent are we physical or mental? All of these questions are still relevant today. We still question the duality of the physical and the spiritual. We still struggle with issues of human guilt.
Dostovesky's novel Crime and Punishment is very relevant today with so many sociopathic individuals who believe, as Raskolnikov did, that they can commit grievous acts with impunity. Time and time again, the newspapers are replete with stories of people such as the main character, who feels himself superior to rules, who feels nothing, at times.
An intensely dramatic and insightful study of the nature of good and evil, Crime and Punishment, the quintessential psychological novel is also the ultimate Russian novel that examines the social conditions of 19th century Russia and its liberal and radical politics.
I think the themes of resurrection and the second chance that Raskolnikov is given are likewise very important, in addition to the psychological issues given above. Unfortunately, individuals who commit horrific crimes are a symptom of all ages, but where this novel is different is that it charts its criminal's coming to terms with his actions and also shows us his life ahead. It is possible to make amends and to move on.
I'd suggest that the continuing relevance of this novel will come from its comment that true guilt is inescapable, regardless of philosophy. It is not "superior" humans that can escape guilt, only crazy people, as suggested by post #2.
The conclusion that his novel points to is also enduring, which is that the truly great people will lift up humanity, not leave it behind.