Acts and Luke were originally anonymous, so actual evidence of authorship does not provide any clue as to their date of authorship, but the second-century Church Fathers attributed the books to Luke, the physician and companion of Paul. This assumption would place Acts within the lifetime of Luke and probably therefore before 70 AD. This is probably the strongest argument for an early date, although there are counter arguments against authorship by Luke or by any other companion of Paul.

An argument from silence, summarised here and here, is that Acts does not mention the outcome of Paul's trial or the destruction of Jerusalem in 70 AD, which is seen as evidence that the book was written before those events. Arguments from silence are not always weak and can be useful in the absence of other evidence, either for or against a case. Ben Witherington asks in The Acts of the Apostles, page 619, whether the silence in regard to Paul's fate means that Luke does not want to portray the untimely demise of his hero, adding that if his intention was not biographical, it would be understandable that Acts ends the way it does.

The Christian Apologetics and Research Ministry, in 'When was the book of Acts written?' expands on some of the above arguments, as well as relying on internal evidence (the 'we' passages) for an early date.

The most common argument I've seen is that Acts omits several important surrounding events, as pointed out by Nathaniel.

Jonathan McLatchie writes at The Christian Apologetics Alliance that

the Acts of the Apostles (which post-dates Luke’s gospel) does not mention the destruction of the temple in AD 70, nor the death of Peter or Paul, nor for that matter the persecution of Christian martyrs under Nero in the 60s or the Great Fire of Rome from which it resulted. If such events had already taken place by the time Luke wrote Acts, one would expect to find a pertaining description. But, instead, Acts leaves us hanging, by ending after Paul has been placed under house-arrest.

J. Warner Wallace at Cold Case Christianity makes the further point that Acts does include some important (and similar) events, so it's not a case of Luke omitting other important events on the grounds that they simply don't fit into his narrative (some emphasis added):

I make a much more elaborate and cumulative case for the early dating of the gospels in Cold Case Christianity, but a portion of this case revolves around Luke’s omission of three important deaths in the Book of Acts: the deaths of Paul, Peter and James. A recent listener to the Stand to Reason “Please Convince Me” Podcast recently wrote: “Firstly, perhaps such historical events were simply beyond the scope of the author of Acts? It has been suggested that the author may have been aware of the aforementioned events, but he chose instead to end his account thematically with the Gospel finally reaching the heart of Gentile civilization, Rome… Is it really viable to suggest that these possibilities are less reasonable than the early dating hypothesis?” One of the evidences in the Book of Acts that makes the omission of Paul, Peter and James’ death so powerful is the inclusion of two other deaths in the narrative: the deaths of Stephen and James, the brother of John:

Acts 7:58-60

When they had driven him out of the city, they began stoning him; and the witnesses laid aside their robes at the feet of da young man named Saul. They went on stoning Stephen as he called on the Lord and said, “Lord Jesus, receive my spirit!” Then falling on his knees, he cried out with a loud voice, “Lord, do not hold this sin against them!” Having said this, he fell asleep.

Acts 12:1-2

Now about that time Herod the king laid hands on some who belonged to the church in order to mistreat them. And he had James the brother of John put to death with a sword.

As important as Stephen and James, the brother of John, were to the early Church, it can hardly be argued that Paul, Peter and James, the three most important Christian leaders of the first century and the primary characters of the Book of Acts narrative, would not be considered important enough to describe their deaths. Is it possible (viable) that Luke “may have been aware of the aforementioned events, but chose instead to end his account thematically with the Gospel finally reaching the heart of Gentile civilization, Rome?” Of course it’s possible, because anything and everything is possible. But it’s not reasonable. And as I’ve described in my book, the difference between what’s possible and what’s reasonable is critically important to jurors. Jurors are instructed that they are not to spend time “speculating” in their jury deliberation about what’s possible, but to consider instead what is reasonable in light of the evidence. The inclusion of the deaths of Stephen and James (the brother of John), make the exclusion of the deaths of Paul, Peter and James (the brother of Jesus) powerful evidences for the early dating of Luke’s work. The most reasonable inference from the omission of these deaths is that Luke wrote his narrative prior to their occurrence.

Acts has traditionally been attributed to Luke, a gentile companion of Paul of Tarsus. This same Luke is likewise believed to have written the third gospel in the NT canon, Luke. Whether these traditions are true or not is anyone's guess, though there is much evidence that Luke was actually Lucius of Cyrene, a Jewish Christian mentioned in Acts 13:1. Since Nathaniel has already provided an excellent answer, this answer will only deal with refuting common criticism of an early date for Acts.

Markian Priority

It has been assumed by many scholars that Mark was the first to write a gospel of Messiah, and that both Matthew and Luke used Mark to write their own accounts. This is highly debatable, and several members of the early church say that Matthew was the first to write a gospel, originally in the Hebrew language. Nevertheless, whether Matthew, Mark, Luke, or John wrote first is irrelevant because if Mark wrote in 36 AD and Luke wrote afterwards, this would not determine anything about the date of Acts. The single strongest argument that Mark was written in 70 AD is based on Mark 13:1

"And as he went out of the temple, one of his disciples saith unto him, Master, see what manner of stones and what buildings are here!

And Jesus answering said unto him, Seest thou these great buildings? there shall not be left one stone upon another, that shall not be thrown down." Mark 13:1-2

The secular argument is that Mark mentions the destruction of the temple, so therefore he must have written his gospel after the temple was destroyed in 70 AD. This is because secular scholars do not believe in prophecy, and therefore find it impossible that Yeshua could predict the destruction of the temple 35-40 years before it's collapse. John A.T. Robinson has written a very thorough book called Redating the New Testament in defense of Yeshua's prediction, showing that neither Mark nor Matthew use language that would be associated with ex eventu prophecies (prophecies written after the prophecied event happened):

"If one really wants to see what ex eventu prophecy looks like, one should turn to the so-called Sibylline Oracles (4.125-7):

And a Roman leader shall come to Syria, who shall burn down Solyma's [Jerusalem's] temple with fire, and therewith slay many men, and shall waste the great land of the Jews with its broad way. [Tr. R. H. Charles, The Apocrypha and Pseudepigrapha of the Old Testament II, Oxford 1913,395.]

It is precisely such detail that one does not get in the New Testament."

So, if one does not believe in prophecy, then Mark was written in 70 AD and Yeshua never predicted the destruction of the temple. If one does believe in prophecy, then Yeshua correctly predicted the destruction of the temple, and we still have no idea when Mark or Luke was written.

Josephus

The second common criticism of an earlier date is that Josephus and Acts seem to contradict each other about whether Theudas or Judas of Galilee led their revolutions first. In Acts, Luke is recording a warning from Gamaliel, in which he says:

"For before these days rose up Theudas, boasting himself to be somebody; to whom a number of men, about four hundred, joined themselves: who was slain; and all, as many as obeyed him, were scattered, and brought to nought.

After this man rose up Judas of Galilee in the days of the taxing, and drew away much people after him: he also perished; and all, even as many as obeyed him, were dispersed." Acts 5:36-37

However, in Josephus' Antiquities, he says that a man named Judas of Galilee rose first, and afterwards a man named Theudas. This would put Theudas' revolt about ten years after Gamaliel said this. There are two objections to this contradiction. The first is whether or not the Theudas in Acts is the same Theudas that Josephus mentions. In his Antiquities, Josephus says:

"Now it came to pass, while Fadus was procurator of Judea, that a certain magician, whose name was Theudas, persuaded a great part of the people to take their effects with them, and follow him to the river Jordan; for he told them he was a prophet, and that he would, by his own command, divide the river, and afford them an easy passage over it; and many were deluded by his words. However, Fadus did not permit them to make any advantage of his wild attempt, but sent a troop of hors*m*n out against them; who, falling upon them unexpectedly, slew many of them, and took many of them alive. They also took Theudas alive, and cut off his head, and carried it to Jerusalem. This was what befell the Jews in the time of Cuspius Fadus's government." Antiquities 20.5.1

It would be a far stretch to say that "four-hundred men" refer to the same group as "a great part of the people". Josephus says that during this time, there were 10,000 disorders in Judea:

Now at this time there were ten thousand other disorders in Judea, which were like tumults, because a great number put themselves into a warlike posture, either out of hopes of gain to themselves, or out of enmity to the Jews." Antiquities 17.10.4

Many people had similar names in those times, so there is no reason why Gamaliel's Theudas should be the same as Josephus. In fact, the Jewish Encyclopedia mentions a different Theudas, that introduced eating a lamb on the eve of Passover to the Romans. This angered the Jews, but this Theudas was always highly respected.

Since Acts and Josephus are the only two historical documents we have that even mention a revolt by a Theudas, it cannot be definite that they both refer to the same man.

The other argument is that Josephus was simply mistaken. It seems unfair that we should attribute a mistake to Acts but not to Josephus, who wrote Antiquities from about 93-97 AD. If Luke was recording the words of Gamaliel, then Gamaliel would have been a first-hand witness of Theudas' rebellion.

Here are a few quotations extracted from the Wikipedia article Historical reliability of the Acts of the Apostles

"Neither position has much of a following today, because of the significant differences between the two works in their accounts of the same events.", Mason, ‘Josephus and the New Testament’, p. 185 (1992)

"After examining the texts myself, I must conclude with the majority of scholars that it is impossible to establish the dependence of Luke-Acts on the Antiquitates.", Sterling, 'Historiography and Self-Definition: Josephus, Luke-Acts, and Apologetic Historiography', Supplements to Novum Testamentum, pp. 365–366 (1992)

"The relationship between Luke and Josephus has produced an abundant literature, which has attempted to show the literary dependence of one on the other. I do not believe that any such dependence can be proved.", Marguerat, 'The First Christian Historian: writing the "Acts of the Apostles"', p. 79 (2002). Cambridge University Press.

Conclusion

As Nathaniel shows, there is overwhelming evidence that Acts was written prior to 70 AD. Any refutation to this is based solely on conjecture and a secular opinion of Mark's gospel. There is no reason that Acts should not be accepted as a reliable and historical account of the events proceeding Yeshua's resurrection.