Peter gave the qualifications for an apostle when they meet to name a replacement for Judas.
Acts 1:21 Wherefore of these men which have companied with us all the time that the Lord Jesus went in and out among us, 22 Beginning from the baptism of John, unto that same day that he was taken up from us, must one be ordained to be a witness with us of his resurrection.
The term apostle was also applied to Paul, who, although he did not meet the above qualifications, was personally set apart as an apostle to the Gentiles by Christ himself. The apostolic age ended when the last (John I think) one died.
With them doctrinal authority also died.