Here are some conciliar and magisterial statements about Purgatorio:
We believe...that the souls, by the purifying compensation are purged after death.
Session IV, Council of Lyons (1274)
By this Constitution which is to remain in force for ever, we, with apostolic authority, define the following: According to the general disposition of God, the souls of all the saints who departed from this world before the passion of our Lord Jesus Christ and also of the holy apostles, martyrs, confessors, virgins and other faithful who died after receiving the holy baptism of Christ- provided they were not in need of any purification when they died, or will not be in need of any when they die in the future, or else, if they then needed or will need some purification, after they have been purified after death-and again the souls of children who have been reborn by the same baptism of Christ or will be when baptism is conferred on them, if they die before attaining the use of free will: all these souls, immediately after death and, in the case of those in need of purification, after the purification mentioned above, since the ascension of our Lord and Saviour Jesus Christ into heaven, already before they take up their bodies again and before the general judgment, have been, are and will be with Christ in heaven, in the heavenly kingdom and paradise, joined to the company of the holy angels.
Pope Benedict XII, Benedictus Deus (1336)
It is likewise defined, that, if those truly penitent have departed in the love of God, before they have made satisfaction by worthy fruits of penance for sins of commission and omission, the souls of these are cleansed after death by purgatorial punishments...
Laetentur Coeli, Council of Florence (1439)
If anyone says that after the reception of the grace of justification the guilt is so remitted and the debt of eternal punishment so blotted out to every repentant sinner, that no debt of temporal punishment remains to be discharged either in this world or in purgatory before the gates of heaven can be opened, let him be anathema.
Session VI, Canon XXX, Council of Trent (1545-1563)
Wherefore, according to the tradition of the Apostles, it is rightly offered not only for the sins, punishments, satisfactions and other necessities of the faithful who are living, but also for those departed in Christ but not yet fully purified.
Session XXII, Chapter II: "The Sacrifice of the Mass is Propitiatory Both for the Living and the Dead," Council of Trent (1545-1563)
Whereas the Catholic Church, instructed by the Holy Ghost, has from the Sacred Scriptures and the ancient tradition of the Fathers taught in Councils and very recently in this Ecumenical synod that there is a purgatory, and that the souls therein are helped by the suffrages of the faithful, but principally by the acceptable Sacrifice of the Altar; the Holy Synod enjoins on the Bishops that they diligently endeavor to have the sound doctrine of the Fathers in Councils regarding purgatory everywhere taught and preached, held and believed by the faithful.
Session XXV, Council of Trent (1545-1563)
1030 All who die in God's grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven.
1031 The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned. The Church formulated her doctrine of faith on Purgatory especially at the Councils of Florence and Trent. The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire:
As for certain lesser faults, we must believe that, before the Final Judgment, there is a purifying fire. He who is truth says that whoever utters blasphemy against the Holy Spirit will be pardoned neither in this age nor in the age to come. From this sentence we understand that certain offenses can be forgiven in this age, but certain others in the age to come. (St. Gregory the Great, Dial. 4,39:PL 77,396; cf. Mt 12:31.)
This teaching is also based on the practice of prayer for the dead, already mentioned in Sacred Scripture: "Therefore [Judas Maccabeus] made atonement for the dead, that they might be delivered from their sin." From the beginning the Church has honored the memory of the dead and offered prayers in suffrage for them, above all the Eucharistic sacrifice, so that, thus purified, they may attain the beatific vision of God. The Church also commends almsgiving, indulgences, and works of penance undertaken on behalf of the dead:
Let us help and commemorate them. If Job's sons were purified by their father's sacrifice, why would we doubt that our offerings for the dead bring them some consolation? Let us not hesitate to help those who have died and to offer our prayers for them. (611 St. John Chrysostom, Hom. in 1 Cor. 41,5:PG 61,361; cf. Job 1:5.)
Catechism of the Catholic Church 1030-1032
1472 To understand this doctrine and practice of the Church, it is necessary to understand that sin has a double consequence. Grave sin deprives us of communion with God and therefore makes us incapable of eternal life, the privation of which is called the "eternal punishment" of sin. On the other hand every sin, even venial, entails an unhealthy attachment to creatures, which must be purified either here on earth, or after death in the state called Purgatory. This purification frees one from what is called the "temporal punishment" of sin. These two punishments must not be conceived of as a kind of vengeance inflicted by God from without, but as following from the very nature of sin. A conversion which proceeds from a fervent charity can attain the complete purification of the sinner in such a way that no punishment would remain.
Catechism of the Catholic Church 1472