Cessationism is the belief that the miracles performed Jesus and the apostles occurred solely to attest to the authority and inspiration of the New Testament, and that spiritual gifts ceased after the writing of the NT was finished.
As writers such as Jack Deere have argued, this is a position with no biblical foundation. He shows that miracles authentic the character of Jesus (John 3.2; 9:32-33; etc.) and the message of Jesus (Mk 16.20) but have nothing to say about the apostles! His argument makes convincing reading in Surprised by the Power of the Spirit.
It also has a problem with the historical record. That record shows clearly that the early church was quite active in the charismatic gifts at least through 200 AD. There was a decline in the 3rd century, and then again became active.
The early church experienced martyrdom and persecution, resisted Gnostics and Arians and doctrinal disputes, and established which of the books we accept as the New Testament canon. Tertullian scoffed at those who tried to translate the gospel into the categories of Greek philosophy. Origen of Alexandria nearly single-handedly invented the systematic study of the Bible. Irenaeus defended the faith against a host of heresies and spoke of the Work of Christ in illuminating new ways. Cyprian insisted on the unity of the church and its necessity for salvation.
And this same early church and same church fathers from the 100s to well beyond the 200s AD — Tertullian, Cyprian, Irenaeus, and many more — experienced and wrote about miracles of healing, prophecy, and exorcism as everyday occurrences in the church.
Tertullian is typical when he says “God everywhere manifests signs of his own power—to his own people for their comfort, to strangers for a testimony unto them” (Tertullian, A Treatise on the Soul).
In other words, the early Christian church was a charismatic church. A community of Christians in the 100s and 200s continued to experience the charismata, the spiritual gifts, described by the Apostle Paul in 1 Corinthians.
The influential bishop of Carthage, Cyprian, for example, says:
In Christianity there is conferred (upon pure chastity, upon a pure mind, upon pure speech) the gift of healing the sick by rendering poisonous potions harmless, by restoring the deranged to health, and thus purifying them from ignominious pains, by commanding peace for the hostile, rest for the violent, and gentleness for the unruly, by forcing—under stress of threats and invective—a confession from unclean and roving spirits who have come to dwell within mankind, by roughly ordering them out, and stretching them out with struggles, howls, and groans, as their sufferings on the rack, by lashing them with scourges, and burning them with fire. This is what goes on, though no one sees it; the punishments are hidden, but the penalty is open. Thus what we have already begun to be, that is, the Spirit we have received, comes into its kingdom.
The 19th-century church historian Adolf Harnack, in his Mission and Expansion of Christianity, categorises the charismatic activities of the 2nd- and 3rd-century church —and the list is impressive:
(1) God speaks to the missionaries in visions, dreams, and ecstasy, revealing to them affairs of moment and also trifles, controlling their plans, pointing out the roads on which they are to travel, the cities where they are to stay, and the persons whom they are to visit. Visions occur especially after a martyrdom, the dead martyr appearing to his friends during the weeks that immediately follow his death, as in the case of Potamiaena (Eus., H.E., vi. 5), or of Cyprian, or of many others.
It was by means of dreams that Arnobius (Jerome, Chron., p. 326) and others were converted. Even in the middle of the third century, the two great bishops Dionysius and Cyprian’ were both visionaries. . . .
(2) At the missionary addresses of the apostles or evangelist, or at the services of the churches which they founded, sudden movements of rapture are experienced, many of them being simultaneous seizures; these are either full of terror and dismay, convulsing the whole spiritual life, or exultant outbursts of a joy that sees heaven opened to its eyes. The simple question, “What must I do to be saved?” also bursts upon the mind with an elemental force.
(3) Some are inspired who have power to clothe their experience in words-prophets to explain the past, to interpret and to fathom the present, and to foretell the future. Their prophecies relate to the general course of history, but also to the fortunes of individuals, to what individuals are to do or leave undone.
(4) Brethren are inspired with the impulse to improvise prayers and hymns and psalms.
(5) Others are so filled with the Spirit that they lose consciousness and break out in stammering speech and cries, or in unintelligible utterances—which can be interpreted, however, by those who have the gift.
(6) Into the hands of others, again, the Spirit slips a pen, either in an ecstasy or in exalted moments of spiritual tension; they not merely speak but write as they are bidden.
(7) Sick persons are brought and healed by the missionaries, or by brethren who have been but recently awakened; wild paroxysms of terror before God’s presence are also soothed, and in the name of Jesus demons are cast out.
(8) The Spirit impels men to an immense variety of extraordinary actions—to symbolic actions which are meant to reveal some mystery or to give some directions for life, as well as to deeds of heroism.
(9) Some perceive the presence of the Spirit with every sense; they see its brilliant light, they hear its voice, they smell the fragrance of immortality and taste its sweetness. Nay more; they see celestial persons with their own eyes, see them and also hear them; they peer into what is hidden or distant or to come; they are even rapt into the world to come, into heaven itself, where they listen to “words that cannot be uttered.”
One of the best more academic articles on the subject is by Nigel Scotland: http://www.earlychurch.org.uk/pdf/ejt/signs_scotland.pdf