The Gospel of Luke is addressed to Theophilus with the expression: “most excellent Theophilus” (Luke 1:3). This phrasing indicates that Theophilus was a Roman official, and not merely a friend or associate of Luke’s. Both the Gospel of Luke and Acts of the Apostles are addressed to this Theophilus. Yet he could not have been very high up in the Roman government because nothing is known about him from other historical evidence. Why then would Luke address two works of such length and importance (both the Gospel and Acts were fairly lengthy works for that time period) to Theophilus?
Perhaps this Roman official expressed an interest in learning about Christ and about the Apostles. He may have been interested in converting to the Faith. Even so, Luke would only address both of these important works to a minor Roman official if Luke were living in the area ruled by that official when he was writing. Since Acts of the Apostles was most likely written sometime during the two years after Paul arrived for his appeal to the emperor, the Gospel of Luke was also written about that same time. At other times, Luke had been traveling with Paul and would not have been in one place, where Theophilus held office, for the length of time needed to write his Gospel. Thus the Gospel of Luke was written early in that two-year period of time and Acts of the Apostles was written in the latter part of those two years.
Luke’s Gospel and Acts are both addressed to the same Roman official, Theophilus. Luke uses the formal term “most excellent” to address Theophilus in his Gospel (Lk 1:3). This language was commonly used in addressing Roman officials. Examples of this same expression are found in Acts, where it is used to address both Felix and Festus (Acts 24:2; 26:25). But, at the beginning of Acts, Luke no longer calls Theophilus “most excellent Theophilus,” instead he calls him “O Theophilus” (Lk 1:3; Acts 1:1). This indicates that Theophilus was no longer in office by the time that Luke was writing, or had completed writing, Acts. Yet Luke still addresses Acts to Theophilus. He must still have been living in the same area and had continued respect and (unofficial) authority at that time. Minor Roman officials often held office for only a year. For example, Gallio (mentioned above) was proconsul of Achaia, probably for only one year. Theophilus could have been proconsul of Achaia, the region in which Jerome tells us Luke was living when he wrote the Gospel, or he could have held some other office within the region of Achaia or Boeotia, (but there is no direct historical evidence as to where Theophilus held office).
(from the book, Important Dates in the Lives of Jesus and Mary)