The Bible is very clear that God is a God of Justice - Elohim Mishpat. In fact, one of His Old Testament names is El Neqamah - the God who avenges or vindicates me. He's not an "avenger" in a petty, payback way, but rather in the sense of righting wrongs, redressing the balance and, where necessary, judging wicked peoples like the Philistines or the Babylonians.
So, probably the most important thing for us to apply to our own lives is that we should never seek revenge on those who have hurt, betrayed or ill-treated us, because only God knows their hearts and He is perfectly just. We should leave things in God's hands, as it says in Psalm 94: 1 or in Romans 12: 17 - 19.
A closely related understanding is that God is a God who vindicates us. Although mishpat is often translated "judgement" in older Bible versions, many versions from the past eighty years use the word vindicate. Vindicate comes from the Latin verb vindicare, meaning to set free, to protect or to avenge. God sets the record straight when our reputation has been damaged, when we've been slandered or falsely accused. It may not happen immediately, but God will be faithful to do it at the right time.
Vindication is promised several times in the Old Testament (in Psalm 24: 5, Psalm 37: 6, and Isaiah 62: 1 - 2, for example) but perhaps the best known is the promise in Isaiah 54: 17. "No weapon formed against you will prosper, and every tongue that accuses you will be silenced. This is the heritage of the servants of the Lord and this is their vindication from me, declares the Lord." Here too, the understanding is that we shouldn't rush to defend our own reputation, but can leave our vindication in God's hands.
The God who judges, avenges, vindicates and brings justice is not just an Old Testament concept; we find it also in the New Testament, for example in the parable that Jesus told about an unjust earthly judge. (Luke 18: 7 - 8)
As Abraham proclaimed back in Genesis 18: 25, the Judge (shaphat) of all the earth will always do what is right and just (mishpat).
