From the Acts of Peter and Paul, an early document; though not canonical, it contains Church tradition:
Then Nero, having summoned Agrippa the proprætor, said to him: It is necessary that men introducing mischievous religious observances should die. Wherefore I order them to take iron clubs, and to be killed in the sea-fight. Agrippa the proprætor said: Most sacred emperor, what you have ordered is not fitting for these men, since Paul seems innocent beside Peter. Nero said: By what fate, then, shall they die? Agrippa answered and said: As seems to me, it is just that Paul's head should be cut off, and that Peter should be raised on a cross as the cause of the murder. Nero said: You have most excellently judged.
Then both Peter and Paul were led away from the presence of Nero. And Paul was beheaded on the Ostesian road.
And Peter, having come to the cross, said: Since my Lord Jesus Christ, who came down from the heaven upon the earth, was raised upon the cross upright, and He has deigned to call to heaven me, who am of the earth, my cross ought to be fixed head down most, so as to direct my feet towards heaven; for I am not worthy to be crucified like my Lord. Then, having reversed the cross, they nailed his feet up.
And the multitude was assembled reviling Cæsar, and wishing to kill him. But Peter restrained them, saying: A few days ago, being exhorted by the brethren, I was going away; and my Lord Jesus Christ met me, and having adored Him, I said, Lord, whither are You going? And He said to me, I am going to Rome to be crucified. And I said to Him, Lord, were You not crucified once for all? And the Lord answering, said, I saw you fleeing from death, and I wish to be crucified instead of you. And I said, Lord, I go; I fulfil Your command. And He said to me, Fear not, for I am with you. On this account, then, children, do not hinder my going; for already my feet are going on the road to heaven. Do not grieve, therefore, but rather rejoice with me, for today I receive the fruit of my labours. And thus speaking, he said: I thank You, good Shepherd, that the sheep which You have entrusted to me, sympathize with me; I ask, then, that with me they may have a part in Your kingdom. And having thus spoken, he gave up the ghost.
And immediately there appeared men glorious and strange in appearance; and they said: We are here, on account of the holy and chief apostles, from Jerusalem. And they, along with Marcellus, an illustrious man, who, having left Simon, had believed in Peter, took up his body secretly, and put it under the terebinth near the place for the exhibition of sea-fights in the place called the Vatican...
And the consummation of the holy glorious Apostles Peter and Paul was on the 29th of the month of June— in Christ Jesus our Lord, to whom be glory and strength.