First of all, The Iliad is extremely important as a historical document and a lens through which we can examine the ancient past. Through this epic, we can gain insights into the religious life of the Ancient Greeks, as well as ancient warfare and culture. On these grounds alone, it is a profoundly important piece of our cultural heritage.
Even beyond this, however, The Iliad stands as a deeply influential and powerful work of literature. It contains themes on military glory (as well as on the costs of war) as well as striking depictions of pride and arrogance (notably expressed in the conflict between Agamemnon and Achilles, both of whom are driven by pride and ego). One sees the interplay of vividly rendered personalities, both mortal and immortal, and the ways in which humans and gods ultimately shape and influence one another (consider the ways in which the Trojan War causes turmoil and division among the gods, many of whom themselves hold a stake in that conflict). As a poem, The Iliad is rich in dramatic weight. From that perspective, one can see how it has endured for so long and why it has held such a deep and lasting influence. It is one of the great masterworks in all of literature.
The topic of the modern-day relevance of The Iliad offers scholars many different facets to explore. War, suffering, heroism, and masculinity are all themes from The Iliad that many would agree are contemporary hot topics.
The Iliad explores the themes of suffering and war in exceptional detail, themes which are inextricably tied with the act of telling stories about heroes. Heroes are individuals who inspire others, and no matter what time in history a reader of The Iliad lives, this kind of inspiration is important. Definitions of heroic behavior may change, but culture's need for heroes is static.
As well, conflicts between groups of humans have always existed in history, and they exist today as global newspapers report daily. The Iliad demonstrates to some that the impulse to go to war is perhaps a human one that cannot be mollified no matter how convincing a pacifist argument might sound. To others, war is one manifestation of masculine anger and aggression, and other scholars say that these emotions need release; such a release will happen, whether societies want this release or not. No matter how an individual feels about war, the reality is that war is a facet of society that seems ineradicable, which is one reason why The Iliad still resonates today.
In The Iliad, the important people start wars and have starring roles in these conflicts. Meanwhile, however, ordinary people go on with their lives as war occurs.
The shield of Achilles shows that life then, as now, consists of more than great warriors or great armies meeting to fight on the battlefield. People still plant and harvest crops. People still find time for worship, and people still create opportunities for song and dance and other festivities. While the battle is raging in one place, ordinary people somewhere else are going about their daily lives.
Homer shows through the shield of Achilles that while war is important, so is ordinary life. He also shows that it is this ordinary life that the warriors are supposed to be fighting to protect. Today too, wars are meant to defend ordinary life. Further, today's everyday people, while battles wage, go on with their existences.
The poem also celebrates doing one's duty, being brave, and putting the needs of the group ahead of personal desires, values that remain important. Heroes then and now are imperfect humans, not saints, something Homer is at pains to show through characters like Achilles.
Now that is a good question. The Iliad is a classic and the reason it is so is because it is so applicable to all of us. The Iliad has many themes that are universal. Moreover, Homer is able to look at these themes in such great depth that it makes us think. Let me give you a few examples. The anger of Achilles is something that we can all relate to, because we all get angry at times. More importantly, what happens when groups or nations feel offended and their honor is hurt? There can be a war on account of great anger. Honor and anger are a few themes that the Iliad focuses on.
Another theme of the epic is war. This theme is always applicable. We are in war now and we probably will be for the rest of human existence. It has been that way and there is no indications of change.
Apart from the idea of themes, we can say that by studying the Iliad we learn a little more about ourselves. Why? Because the Iliad in many ways has shaped the Western way of thinking.