Substitutionary atonement, however it is nuanced with terms like satisfaction, propitiation, penal substitution, et al., is unpopular today. Instead of substitution, scholars have proposed three big substitutionary explanations (sorry for the punning).
Simon Gathercole examines these alternatives in Defending Substitution: An Essay on Atonement in Paul.
In Germany, Stellvertretung in the sense of “place taking.” Gathercole focuses on two scholars, Hartmut Gese and Otfried Hofius. Here are the main lines of thinking of Gese in his study of OT atonement/sacrifice/Day of Atonement:
[The] people as a whole are represented by the priest;
the priest establishes a connection with the animal by laying hands on it;
the animal therefore carries the identity of the priest (and so of the people as a whole);
the sacrifice of the animal enacts the human forfeiting of life;
the life of the animal (and therefore of priest and people) is carried in the blood;
the blood is sprinkled in the Holy of Holies;
the blood thereby comes into contact with God;
ergo, since the blood of the animal comes into contact with God, so do the priest and the people, whose identity is carried by the animal (33).
This is not substitution but inclusive place-taking. The priest identifies with the animal by placing hands on it and the animal takes the priest with it into death, while the blood takes the priest/people into the presence of God. Hofius focuses on the whole person and sin being together so that sin cannot be removed from the person (Kantian idea). Again, his emphasis is on inclusive place-taking.
Gathercole finds four problems: (1) absence of laying on of hands in Lev 16; (2) the scapegoat is substitutionary, as is clear in Lev 16:21-22; (3) too much Kant; (4) sins in the plural is not dealt with.
In England, Morna Hooker’s “interchange” (not “exchange”) theory.
In brief, in this conception of the atonement, it is not that Jesus swaps places with his people-in this death on the cross. Rather, he goes to the place where they are and takes them from there to salvation (38-39).
In short, “Christ identified with the human condition in order that we might be identified with his (39).
Hooker’s theory is formed against exchange and substitution and here are her major elements:
The problem of the human condition is sin and death—that is, the sinful condition of Adamic humanity is the position from which people need to be rescued.
Christ then enters into this condition.
Christ thereby unites himself to us, but humans also must identify with him.
Having been united with him in his death, we pass out of death and into resurrection life with him (receiving the declaration “not guilty” as he did).
We now continue to live with him, and in him we are formed increasingly into his likeness (39-40).
Again, Gathercole’s appreciations are met with some disagreements: (1) he does not agree with her that Paul actually criticizes substitution; (2) he is not convinced this theory appreciates the achievements of the cross enough; (3) what about individual sins?
In the USA, the apocalyptic deliverance model. [SMcK: this approach offers an alternative narrative/story to the whole mission of God in this world.]
This is in conscious and direct contrast to forensic and substitutionary models and its Two Ways: either obey the Law and find salvation or disobey it and find damnation. Here is J. Louis Martyn:
Anti-God powers have managed to commence their own rule over the world, leading human beings into idolatry and thus into slavery, producing a wrong situation that was not intended by God and that will not be long tolerated by him. For in his own time, God will inaugurate a victorious and liberating apocalyptic war against these evil powers, delivering his elect from their grasp and thus making right that which has gone wrong because of the powers’ malignant machinations. This kind of apocalyptic eschatology is fundamental to Paul’s letters.
[In brief, from Martyn:] the human plight consists fundamentally of enslavement to supra-human powers; and God’s redemptive act is his deed of liberation (44).
Very much at work in the apocalyptic model, and Gathercole mentions Martyn, de Boer, Gaventa, and D. Campbell, is upper case Sin and Law and less time for sin as sins.
In contrast to the drama with three actors [God, humans, Christ], one of which is “Christ as the substitutionary sacrifice,” there are four: the introduction of the cosmic power of Sin or the enslaving power of the Law fundamentally changes the landscape and the role of Christ, who (instead?) “comes to embody the enslaving curse” (45).
Gathercole responds: (1) he thinks it works better for Galatians and not for Romans, esp for 1-4 and it does not work for 1 Thess; (2) the plight is reduced too much to victim of cosmic powers and there is an absence of guilt (Rom 1-3); (3) How does this model explain how it works?: (4) what about sins?
Gathercole: “The problem with such models is that, again, if they are seen as dominant, they neglect a crucial factor in Paul’s conception of the atonement, that is, that Paul sees Christ’s death as dealing with sins plural. Sins—individual infractions of the divine will—are frequently mentioned in Paul, and yet one finds them frequently marginalized in scholarship” (48).
The contention that Paul likes Sin and not sins is wrong, and Gathercole compellingly proves this.
The prejudice, therefore, that Paul is not interested in sins (plural) or acts of transgression, however expressed, is a mistaken one. As a result, to explicate the atonement overridingly in terms of victory over Sin as a power is one-sided. Paul frequently refers to the human plight in terms of sins, transgressions, and trespasses, and so it is no surprise to see reference to Christ’s death as dealing with these—even summarizing his gospel this way (50).
Another theory is that Sin is Paul and “sins” pre-Pauline tradition in Paul’s letters. But proving what is pre-Pauline (apocalyptic theory often takes Gal 1:4 that way) is more than difficult and rarely compelling. And, so what if it is pre-Pauline: if Paul picks it up it becomes his, too. Eg., 1 Cor 15:3-8.
Representation and apocalyptic theories often focus on Sin and downplay sins.