Penal Substitutionary Atonement (PSA) is one of many atonement theories, and is the most commonly held theory among evangelicals today. Sadly, Penal Substitutionary Atonement is wrongly assumed to be the only biblical view of atonement, and in the most egregious cases, it is tragically identified with the gospel, and anachronistically imposed on Church history, as if it were the one and only atonement theory since the dawn of time. Penal Substitutionary Atonement was developed in the post-reformation period, and is a modification of Anselm's Satisfaction atonement theory found in his works, such as in Cur Deus Homo? 

Penal Substitutionary Atonement is similar to Anselm's Satisfaction theory, in that they both indicate God that sin is a crime against God, and God is the one who must be placated alone; the ancient church generally held to a ransom theory, or Christus Victor, and similar atonement theories that taught that a sin places a person into the bondage of Satan, and that Satan must be placated, deceived or conquered in order to have their sins atoned and become reconciled.

Penal Substitutionary Atonement's development is sometimes attributed to the reformers, but it is debatable whether John Calvin, Martin Luther, and other initial reformed held to a proto-Penal Substitutionary Atonement theory or not. PSA adds a distinctive violent, judicial element that is not present in Anselm's Satisfaction theory, in that it argues that God's wrath is directed towards sinners, and that each sin must be repaid quid pro quo, and therefore the penalty that each sinner owes to God is paid by Jesus enduring the wrath of God for them on the cross, to appease the wrath of God on their behalf. Perhaps the most famous proponent is Jonathan Edwards, exemplified by his book Sinners in the Hands of an Angry God. 

Penal Substitution theory is therefore deeply flawed and presents an insurmountable theodicy problem because it teaches that God is complicit with violence, and places violence at the center of God's interaction with humanity, in a way that cannot be reconciled with the biblical statements that God is love (1 John 4:8). God is described primarily as an entity of wrath (not love), that may only be expiated through the human sacrifice of God's son. And it is alarming how similar PSA resembles the child sacrifice and molestation to Chemosh, Molech and other pagan religions that was used effectively against Israel in the bible (c.f. 2 Kings 3:27). PSA also has alarming similarities to the ancient heresy of Marcion of Sinope, who described the God of the Old Testament as an evil god that is defeated by Jesus who is the good God of the New Testament. Proponents of Penal Substitution deny that these are valid comparisons, but they are objections that are constantly raised that are not sufficiently answered. 

Eberhard Jüngel's criticism of Penal Substitutionary Atonement 

Eberhard Jüngel provides a helpful correction to Penal Substitutionary Atonement that is consistent with Karl Barth's doctrine of reconciliation, and Moltmann's passibility of God in his book, Death: The Riddle and the Mystery. Jüngel argues that Jesus alone did not endure the cross to placate the wrath of God the Father, but instead the divine impassibility of God is surrendered, such that God in Jesus was crucified for us on the cross. Therefore, Jüngel removes the violence and theodicy issues from Penal Substitution by explaining that God has been crucified on the cross for us, in a similar way to Jürgen Moltmann's The Crucified God and still retains the judicial element the reconciles each individual sin judicially as found in Karl Barth's "The Judge Judged in Our Place" (§59.2) in his Church Dogmatics IV/1 The Doctrine of Reconciliation. 

There is one traditional view of the death of Jesus Christ which must be expressly rejected as a misunderstanding. Based upon certain formulations to be found in the New Testament it is the view that the death of Jesus is to be understood as a human sacrifice to the God of wrath, as an offering intended to appease the wrathful God, who 'like the raging sea seeks out his victim'. No! If we are to speak of any offering at all in connection with the death of Jesus then it must be of the offering of divine otherness, of the surrender of the divine impassibility and absoluteness; in brief, of the sacrifice and surrender of the sheer distinctiveness of God over against his sinful creature. This is what is involved when we speak of God's identification with the dead Jesus. It was not that God let himself be reconciled through the death of Jesus. Enmeshed in its alienation from him, it was he who reconciliation which he has effected (2 Cor 5:20). [1]

Martin Luther was the one who famously called Jesus the "crucified God" (long before Moltmann), and in this same section, Jüngel provides commentary on another famous statement by Luther that Jesus on the cross was the greatest sinner of all. Jüngel argues that Jesus has taken upon the sins of the world in the cross, but he has not done this against God, but God in Jesus has been crucified as the great sinner, and that God now bares the marks of sins. So Jüngel claims Luther as his alley against the proponents of the angry God of wrath presented in Penal Substitutionary Atonement.

To speak to man of the reconciliation which God has brought about means that his eyes should be opened to the fact that man's godlessness and guilt leave their marks upon God himself. In his concerned involvement with the sinner who must suffer the curse of death, God is also involved with man in his godlessness and guilt. It is by taking this godlessness and guilt upon himself that he opposes and contradicts them. Whenever we speak of God's identification with the dead Jesus this is what we mean. Luther expressed this as follows: on the cross, Christ is now 'the greatest robber of all, the greatest murder, adulterer and thief; the greatest desecrator of temples and blasphemer; the world has seen non greater than this.' However, it is only when the believer can rejoice in his need to perceive the marks which his own godlessness has left upon God himself that this statement will be seen to express the very essence of Christian faith. The marks which man's godlessness has left upon God himself are the visible signs of reconciliation. In terms of the language of the doctrine of justification, this would be the most concise way of expressing the meaning of the death of Jesus. [2] 

Sources:

1. Eberhard Jüngel, Death: The Riddle and the Mystery, trans. Iain & Ute Nicol, (Westminster Press, 1975). p. 113.

2. Ibid. pp. 113-4.