It is Catholic doctrine that the Pope, in virtue of his supreme vicarious power, can dissolve for the good of souls any natural, i.e. any non-sacramental, bond of marriage. A non-sacramental bond is one in which neither of the parties, or only one of the parties, is validly baptized as a Christian. The Pope can also dissolve a sacramental marriage [where both parties are validly baptized Christians] which has not been consummated. A dissolution, as opposed to a declaration of nullity, presupposes that the marriage bond is valid and that there are circumstances for the good of souls which justify the intervention of ecclesiastical authority to dissolve that bond. It is important to remember the difference between dissolution, which requires the active intervention of the executive power of the Church, and marriage nullity which is a judicial declaration that a marriage never existed in the first place. Unlike divorce, which claims to break any bond of marriage merely by the will of the parties and according to the criteria of civil divorce law, cases of dissolution can only be granted when strict conditions are in place. Moreover, the Pope cannot dissolve any sacramental marriage which has been ratified [i.e. in which the consent of both parties is valid] and consummated.

Cases of dissolution have a long and complex history and the Church makes reference to their usage even in Apostolic times, e.g. St. Paul (1 Corinthians 7, 12-16).

Cases of dissolution include the “Pauline Privilege” and the “Dissolution in Favour of the Faith.” Also included under the category of dissolution are the less common cases of “non-consummation”.

The "Pauline Privilege" is restricted to the situation in which both parties to a valid marriage are unbaptised. If one party later converts and receives Christian baptism, he or she obtains the right to remarry if the unbelieving spouse refuses to “cohabit peacefully." What does that mean? It means that the unbelieving party hinders the now baptised party in the practice of his/her new faith. That said, it is only if the unbeliever morally and/or physically separates him/herself from the newly believing party that the Pauline Privilege can be invoked. If all the proper conditions are met, and the converted party does remarry, the prior bond of marriage to the non-believer is dissolved by the law itself at the moment in which the new marriage takes place, without any need to have recourse to the Pope.

"Dissolution [of the marriage bond] in Favour of the Faith", also known as the "Privilege of the Faith", applies to cases of valid non-sacramental marriages, which have ended in civil divorce and in which at least one party is not baptised. One example of a situation which can be submitted for a "Privilege of the Faith" is where a party to the first marriage [whether baptised or not] wishes to marry a Catholic in a second ceremony. Another example could be a non-baptised party who wants to become a Catholic and get married to a third party, whether baptised or not. In these cases, the Pope can dissolve the first non-sacramental marriage, provided all that needs to be proved can be proven.

For the Privilege of the Faith to apply, the petitioning party must be neither the exclusive nor the principal cause of the breakdown of the marriage.

The earlier marriage is dissolved at the moment in which the new marriage is celebrated.

These cases refer to marriages between two baptized parties which were validly contracted but in which no completed act of sexual intercourse took place between the wedded couple. After they have validly exchanged consent, the Church considers the first conjugal act between the spouses as the act of consummation [see "glossary" and the associated link for the definition of consummation in Church doctrine]. Once this has occurred between two baptized spouses, their marriage, if valid, is indissoluble even by the Pope. 

If, however, consummation as understood by the Church does not take place, the marriage can be dissolved by the Pope.

Demonstration of non-consummation must reach "moral certainty" [beyond probable doubt] before the papal dispensation will be given. This can be achieved usually by one of three following ways:

1. Physically: with the assistance of medical experts.
2. Morally: The sworn testimony of parties who knew of the non-consummation at the time when the marriage was not in question, as well as evidence of the credibility of the witnesses, documents and other indications which can give moral certainty.
3. By demonstrating from evidence beyond dispute that the parties did not spend sufficient time together to engage in sexual intercourse: e.g. that they separated immediately after the celebration of the wedding.

While the forms of dissolution of marriage mentioned above only take effect at the moment a party contracts a new marriage, the dissolution due to non-consummation is the immediate effect of the dispensation, even if neither party intends to remarry.