In The Kite Runner, what is the relevance of the issues dealt with by the author to today's world?

The list of issues that Hosseini deals with in The Kite Runner is a long one. And I would say that everyone of them is relevant in today's world.  Let's look at a few. 

First, the novel grapples with ethnic and religious discrimination, as it deals with the treatment of the Hazara, Hassan and Ali's people, who are Shia, by the Pashtun, Amir and Baba's people, who are Sunni.   Ethnic and religious discrimination are a profound problem in today's world, all over the world, with wars being waged every day on the basis of both, and with individual instances of hatred as well, such as the burning down of mosques and Muslims being beaten for simply being Muslim.   

Second, the novel deals with the problem of sibling rivalry, in the form of the rivalry that Amir senses between himself and Hassan, even when he does not yet know that Hassan is in fact his brother. Sibling rivalry is a universal theme in literature, from Cain and Abel to A River Runs Through It, and in life, as anyone who has a sibling can tell you.

Third, the novel deals with the tragedy of the upheaval of a nation that sustains repeated invasions and internal strife, as Afghanistan has. The novel stops well before the most recent invasion by the United States, of course, but the devastation shown in the novel, when Amir returns, is a problem today, in many war-torn countries, not just in Afghanistan.

Fourth, the novel shows the immigrant experience of Amir and Baba, strangers in a new land, trying to navigate in a new culture, with the humiliations that this often entails for immigrants, certainly in the United States and probably in many other countries.  Right now, the world is trying to deal with massive migrations of immigrants, and this novel gives, in my opinion, a great deal of insight into the pain and difficulty of the immigrant experience.

Fifth, and perhaps most universal, the novel is about guilt, atonement, and redemption, which are universal themes that we can all identify with. Who amongst us has not felt guilt? Who amongst us has not tried to atone for a wrongdoing? That is the road to redemption, a long journey for Amir, literally and figuratively, as he admits to himself his wrongdoing and finds his way "to be good again" (2).  

This novel is packed with important themes that reflect universality and timelessness. All matter in today's world. As long as we are human, we need to be reminded of our flaws, hatred and bias, cruelty and envy, the horror of war, our unkindness to strangers, and how we can redeem ourselves from our wrongs.