Catholic
The Catholic Church teaches that by his death on the cross, Jesus makes reparation for our sins and reconciles humanity to God, accomplishing a satisfaction no human could offer. It holds that the fruits of his atoning sacrifice are applied to believers through the sacraments and a life of grace.
Source: CCC 616–618; 1131 ↗
Orthodox
The Orthodox Church teaches that Christ's death and resurrection defeated death itself, ransoming humanity from the devil, sin, and corruption. It emphasizes Christ's victory over death and the healing of human nature rather than a payment of penalty owed to the Father's wrath.
Source: The Orthodox Faith, Vol. I — Redemption ↗
Lutheran
Lutherans confess that Christ's innocent suffering and death were a vicarious satisfaction for the sins of the whole world, by which God's wrath was appeased and reconciliation accomplished. They hold that this atonement is received through faith alone, not by human works.
Source: Augsburg Confession, Art. IV; Apology of the Augsburg Confession, Art. IV ↗
Reformed
The Reformed tradition teaches penal substitution: Christ bore the full penalty of God's wrath due to sinners, satisfying divine justice in their place. It holds that his death was a definite, effective atonement for all whom the Father had given him.
Source: Westminster Confession 8.5 ↗
Baptist
Baptists teach that in his substitutionary death on the cross, Christ made provision for the redemption of all people from sin, obtaining eternal redemption for the believer by his own blood. They hold that this atonement is received through personal faith in Jesus Christ as Lord.
Source: BFM 2000, II.B; IV ↗
Methodist
Methodists teach that Christ's death was an atonement for the sins of the whole world — sufficient for all and offered to all, in keeping with God's universal love. They hold that its benefits are received through repentance and faith, enabled by God's prevenient grace.
Source: Articles of Religion, Art. XX; UMC What We Believe ↗