Why does evil exist in the world?
There is divine wisdom behind the existence of what we perceive as 'evil' in the world. We can attribute its existence to a few general reasons.

Evil as a Test
Muslims view life as a test through which human beings strive to come closer to God and develop their relationship with Him by submitting to the divinely ordained way of life. Consequently, any calamity that befalls us is a test for us, and those who patiently persevere through such afflictions and stand strong in their devotion to God are the ones who have succeeded. Likewise, afflictions that befall others in the world around us are also tests for us. Had there been no violence in the world, where would the test be to strive for peace? Had there been no poverty, how would our generosity be tested? Had the world been free on injustice, then how could anyone make noble sacrifices in the cause of justice? Our good deeds bring us closer to God, and these deeds are contingent on the existence of some problem in the world around us. If we patiently persevere through calamities, enjoin the good and forbid the evil, we will be rewarded immensely in the next life. When a person commits a sin or immoral act, they are failing in their test. They are failing to properly use their abilities that God has entrusted them with. The duty to strive against evil in this world is a duty that has been entrusted to human beings as a test for them. Thus, when human beings fail to fulfill their duty, the deficiency is in them and NOT in God.

Example from Qur'an:
Every soul will taste death. And We test you with evil and with good as trial; and to Us you will be returned. [Surah Al-Anbiya, v.35]

Evil as a Reminder
In addition to their function as tests, calamities also serve as reminders for us. They remind us of the temporal nature of the life of this world, and they also remind us of the fact that true happiness and peace only comes in devotion to God alone, and not materialistic pleasures. We are reminded by sad events of the inevitable end of life and we are compelled to reflect on what we have prepared for the next life. We are reminded of the need to return to God's revelation for the promotion of virtue in our world, and we are reminded that abandoning path of God entails the rise of evil. We are reminded that no matter how great we may think ourselves to be, God is the Most Great and He is the Master in control of His servants. In the blink of an eye, He can take away the blessings that He has bestowed one with, so it behooves the true servant of God to be always grateful.

Example from Qur'an:

The story of the man with Two Gardens
And present to them an example of two men: We granted to one of them two gardens of grapevines, and We bordered them with palm trees and placed between them [fields of] crops. Each of the two gardens produced its fruit and did not fall short thereof in anything. And We caused to gush forth within them a river. And he had fruit, so he said to his companion while he was conversing with him, "I am greater than you in wealth and mightier in [numbers of] men." And he entered his garden while he was unjust to himself. He said, "I do not think that this will perish - ever. And I do not think the Hour will occur. And even if I should be brought back to my Lord, I will surely find better than this as a return." His companion said to him while he was conversing with him, "Have you disbelieved in He who created you from dust and then from a sperm-drop and then proportioned you [as] a man? But as for me, He is Allah , my Lord, and I do not associate with my Lord anyone. And why did you, when you entered your garden, not say, 'What Allah willed [has occurred]; there is no power except in Allah '? Although you see me less than you in wealth and children, It may be that my Lord will give me [something] better than your garden and will send upon it a calamity from the sky, and it will become a smooth, dusty ground, Or its water will become sunken [into the earth], so you would never be able to seek it." And his fruits were encompassed [by ruin], so he began to turn his hands about [in dismay] over what he had spent on it, while it had collapsed upon its trellises, and said, "Oh, I wish I had not associated with my Lord anyone." And there was for him no company to aid him other than Allah , nor could he defend himself. [Surah Al-Kahf, vv. 32-43]

Evil as a Punishment
Furthermore, calamities also serve as expiation for any of our misdeeds. The sins that we commit in our lives are often the cause, whether directly or indirectly, for much of the tribulations that may befall us. Any punishment from God in this life will remove a proportional amount of sins from a person, and may often increase a person's reward as well, in this life and/or in the Hereafter. This point is especially important to remember in the case of innocent children or other people who are not morally accountable yet undergo major suffering in this world. Sometimes the suffering may be brought about by other human beings. In such cases it reflects the failure of such people in their accountability to God, and they will not go unpunished for their evil. In other cases, innocents may suffer as the result of some neutral cause like a medical disease. In either event, for those who are suffering they will be recompensed in the next life with a reward that is infinitely greater than the suffering they had to endure in this life.

Example from Qur'an:
And whatever of misfortune befalls you, it is because of what your hands have earned. And He pardons much. [Surah Ash-Shurah, v. 30]

Evil has appeared on land and sea because of what the hands of men have earned (by oppression and evil deeds, etc.), that Allah may make them taste a part of that which they have done, in order that they may return in repentance. [Surah Ar-Rum, v. 41]

Wa aleikom salaam

Nazir