We usually read the position that there were only 12 Apostles + Paul, and that after the Apostolic era, we have no more Apostles at all. Specially I see this among Reformed theologians.
But how do these theologians explain Acts 14:14 and Acts 14:4, where Barnabas (not one of the twelve and, obviously, not Paul) is called an Apostle? (ἀπόστολοι is the same word used in Matthew 10:2, for example)
4 But the multitude of the city was divided: and part held with the Jews, and part with the apostles.
14 But when the apostles Barnabas and Paul heard of this [...]
Note on the verse 4: in the passage's context, we see in verse 1 that only Barnabas and Paul were there at Iconium as Christians in that specific "apostolic" mission, so, when we read verse 4, is clear that "apostles" there are Barnabas and Paul and no one else.
Another interpretation: There is a theological branch that will call them "Apostles of the Holy Spirit", because they were separated by the Holy Spirit in Acts 13:2 pretty much as the twelve were separated by the Lord at Matthew 10:2. Mainly Dispensationalists will hold on this view. And they believe that the Goly Spirit still separating His Apostles today. A good answer could deal with this view too.