The hotly debated question of authorship of the three Pastoral Epistles is described in the article about 1 Timothy on the NT Books page.
From the text of Titus itself, we see that Paul had left Titus in Crete (1:5). This would suggest that Paul and Titus had at some time been together in Crete. He says, "let our people learn to apply themselves to good deeds" (3:14). Since Paul called them "our people", it seems that he and Titus probably founded the church in Crete together. Paul tells Titus to meet him at Nicopolis, because he is spending the winter there (3:12). So the letter to Titus was written on the way to Nicopolis, which is on the west coast of Greece.
Several of the people mentioned in the letter are mentioned elsewhere in the New Testament. Artemas (3:12) and Zenas (3:17) are only mentioned here. Tychicus (3:12) was with Paul on his return to Jerusalem on third missionary journey (Acts 20:4), and served as the “postman” for Paul during Roman imprisonment (Eph 6:21, Col 4:7). Later he was sent by Paul to Ephesus (2 Tim 4:12), perhaps being the postman again. In Titus, Paul is planning to send him to Crete to replace Titus as his representative on the island. This time he is not the postman, as Zenas and Apollos delivered the letter (3:12). Tychicus was one of Paul's trusted co-workers, often travelling with him.
Apollos (3:17) is mentioned a number of times in the New Testament. The first time, he helped by Priscilla and Aquila (Acts 18:24), and then went to Corinth (Acts 19:1), after Paul had founded the church there. It appears that he had a special following in Corinth (1 Cor 1:12, 3:4-6,22, 4:6), and Paul has to defend their working relationship. Later he was urged by Paul to visit Corinth (1 Cor 16:12). He had spent some time with the church in Crete and was there when Titus arrived (Titus 3:13)
Biographical study of Titus
We only know a little about Titus, as he is not mentioned in the book of Acts. All the information we have about him is from brief appearances in Paul’s letters.
Titus was a Gentile whom Paul had probably led to the Lord (Titus 1:4). He was taken by Paul and Barnabas to Jerusalem (Gal 2:3) on Paul's second visit there. Paul took him as an test case of a Gentile convert to the discussion on whether Gentiles should first become Jews, coming under the law and being circumcised. Titus was not compelled to be circumcised.
He was one of Paul's faithful co-workers, who was often given difficult and trusted tasks to do. He appears to be a stronger character than Timothy. We see him sent to Corinth as Paul's representative and trouble shooter, with the difficult job of bringing correction into that church. He also organised and was urged to complete their contribution for the Collection for the Jerusalem saints (2 Cor 8:6).
Titus was involved in the tense situation addressed by Paul in 1 Corinthians and he brought the good news of the Corinthians' positive response to his letter to Paul in Macedonia (2 Cor 7:5-16). He carried the second letter to Corinth, and was called Paul's "partner and fellow worker" in the service of the Corinthian church (2 Cor 8:16-24). Titus, like Paul, would not abuse his privilege by exploiting them to his own advantage (2 Cor 12:17-18).
Paul had high regard for Titus, in his recommendation to the Corinthian church in 2 Corinthians, he describes him as "my brother" (2:13), speaks of Titus' heart going out to Corinthians (7:14) and of Titus having earnest care for church at Corinth (8:16). He also says that Titus served the church without taking advantage of them financially (12:18).
Paul had left Titus on the island of Crete to amend what was defective and to appoint elders (Titus 1:5), and later was sent to Dalmatia (2 Tim 4:10).
Eusebius described Titus as being the bishop of Crete until his old age (Ecclesiastical History 3:4). Some people have suggested that he was Luke's brother.
The Church on the Island of Crete
We do not know how the Gospel first came to the island of Crete. There were Jewish Cretans at Day of Pentecost, who heard the Gospel from Peter (Acts 2:11). On his journey to Rome, Paul’s ship attempted to shelter in Crete, but was swept past in a storm (Acts 27). Paul had left Titus on Crete to amend what was defective and to appoint elders in every town (1:5), and quoted one of the poets who described the Cretans as “always liars, evil beasts, lazy gluttons” (1:12).
When was the church at Crete founded?
It is very difficult to answer these questions, as there is so little information. We have no record of Paul going to Crete, or of a church being planted. It may have been a result of people present in Jerusalem on the Day of Pentecost, but Paul refers to the people as "our" people” (3:14), which points to Paul seeing them as "his" planting.
When was the letter written?
Two suggestions have been put forward: The first suggestion was during Paul’s three months in Greece on the third Missionary Journey (Acts 20:3). Perhaps after visiting Corinth, he went with Titus to Crete (two hundred miles from Corinth, which is four days sailing), founded churches and left Titus to establish the church government. He then returned to Nicopolis and on the way wrote the letter of Titus. He spent the winter there (whatever was left of the three months) then proceeded on to Jerusalem, even though it seems that sailing in winter months was avoided (Acts 27:9-12).
We need to remember that Luke was very selective in what he included in the Book of Acts and could easily have omitted this visit, especially as he has not even mentioned Titus at all. Also, Paul seemed able to accomplish much, even in the briefest of visits, the Thessalonian church may well have been established in 3 weeks (Acts 17:2). However even if we accept that Paul was able to do much in a short time, it seems a few weeks is too short for churches to be started over all the island. From the letter it seems that enough time had elapsed from the planting of the church for false teachers to get in and have effect. Also Tychicus was sent to replace Titus (Titus 3;12), but in Acts 20:4, he was with Paul on his journey to Judea.
The second, and more likely, suggestion is to say that Paul’s and Titus' visit was after Paul's Roman imprisonment (Acts 28), some time after the end of the Book of Acts. This would give enough time, as Paul was not martyred until after the Fire of Rome in A.D. 64, two or three years after the end of his imprisonment in Rome.
In the letter to Titus, we can clearly know that Paul wrote the letter on his way to Nicopolis, but we cannot be sure exactly when. As we study this letter, we will find that this lack of solid information in no way hinders our study.
Problems with false teachers in the Church on Crete
There are two main passages which describe the false teachers and what they were doing in the church (1:10-16, 3:8-11). Paul describes them and their message in very strong language.
Who were they?
The false teachers were local people from Crete who had picked up false doctrines from various sources. They were getting into the various churches and causing havoc. Their motive was money. Their consciences were corrupted and they are described as insubordinate, empty talkers, deceivers, liars, evil beasts, lazy gluttons, unbelieving, detestable, disobedient and unfit for any good work.
What were they teaching?
It is difficult to know exactly what was being taught, but it seems it was a mixture, with a strong influence of Judaism, and some early Gnosticism in it. It seems to have similarities with the problems in Ephesus that Timothy was sent to sort out (1 Tim 1:3), but more Jewish. It seems the reference to "controversies and genealogies" point to it not being pure Judaism, as in Galatians and Acts 15. False teaching in the New Testament seems to be either Jewish legalism, Gnosticism, or various combinations of both.
How did this affect the church?
The greatest problem seemed to be that they were lax and indifferent to their responsibility to live in a godly way. There was a dichotomy between belief and lifestyle. This was because Cretans "naturally" proved to be lazy and this was exaggerated by the false teaching. In Col 2:20-23, Paul addresses a similar situation: how legalism and ascetic practices do not have any value in checking the indulgence of the flesh (also see 1 Tim 4:8). Some may have been abusing the grace of God (3:11ff), while others were taking great care to carry out ritual obedience, while ignoring the areas of character and kindness.