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.