Are we allowed to take revenge on a person who has wronged us?
We shouldn’t be vengeful; if someone who deserves punishment can’t be brought to justice, we should trust in G-d to punish him. See Lev.19:18 (“Do not take vengeance…”) and Deut.32:35 (“Vengeance is Mine…”).
Maimonides in Negative Commandment 304 (derived from Lev.19:18) implies that one shouldn’t take revenge even for injuries, and King David implies something similar in Psalms 7:5. But in fact the example of revenge given by the Talmud (Yoma 23a) and cited by Maimonides in the Mishneh Torah (Traits of Character (Deos) 7:7) is that a person shouldn’t refuse to lend something to someone who had previously refused to lend something to him. In this example, no damage was involved. If someone attacks you, either physically or verbally, you are allowed to retaliate, though if possible you should do it through the legal system. This is explained by Nahmanides in his commentary on Lev.19:18. Biblical cases in which revenge was taken include Gen. Ch. 24, when Dinah was abducted and (probably) raped, and 1 Kings Ch. 2, where the man who cursed David deserved punishment for insulting the king. Perhaps not retaliating for an injury is an ideal, but for most of us, such ideal behavior may be too much to expect.