The problem of evil or "Epicurus paradox" attempts to show that the existence of an omniscient, omnipotent and benevolent God is incompatible with reality, insofar as the world is thought to include evil and misery. The problem has been known since antiquity and has engaged several moral philosophers and theologians ever since.
The problem of evil refers to an argument against the existence of a God that is simultaneously described as omnibenevolent (ethically good) and capable of eradicating evil (omniscient and omnipotent), insofar as evil remains. Worse still, such a God would not have allowed evil to flourish in his creation in the first place. This criticism is mainly geared towards Abrahamic religions, although exceptions occur. Calvinism offers an interesting theodicy in which mankind doesn't have the free will to redeem itself, similar to what Leibniz would later conceive as "the best possible world"; so according to those religious views, God is indeed responsible for what we perceive as evil. "Evil" is most commonly operationalised as suffering. This includes the suffering of innocent (i.e. sinless) children and possibly animals, and whose causes could comprise natural disasters, so as to avoid counterarguments based on retribution from man's misuse of free will (and so as to avoid the whole free will debate as well).
Religious responses to the problem of evil are grouped under the banner of "theodicies". God's Mysteriousness Theodicies A common response is to excuse God by downplaying human capacity to understand and recognize true evil and tell it apart from the greater divine design. Atheists reply back that such an explanation incurs in a confirmation bias whereby the good outcomes (e.g. alleged miracles) are never subject to the same skeptical questioning, rendering it unfalsifiable. Free Will Theodicies Augustine of Hippo adopted a view in which evil is brought to the world during the Fall of Man, who freely chose to live in a state of privation from perfection, and therefore evil is allegedly untraceable to God. Assuming the existence of free will, this still raises the question of why an omniscient God could not anticipate the emergence of evil. Greater Good Theodicies The soul-making or Irenaean theodicy states that inexplicable suffering is preferable to an idyllic world. Suffering allows bliss and spiritual growth to occur in the first place, by creating a contrasting quality. In its most extreme version these theodicies maintain that evil is illusory, analogous to how coldness can be described as an absence of heat. Theologically Deflationary Theodicies Theists may accept that God does not posses all of the traditional divine attributes (omnibenevolence, omnipotence, omniscience, omnipresence), without commitment to the conclusion that no supernatural creator whatsoever exists.
The logical argument from evil in modus tollens form is as follows: If an omnipotent, omniscient and omnibenevolent god exists, then evil does not. Evil does exist. Therefore, such a god does not.