How many apostles are there?

Apostle simply means "sent one".

Certainly, there were twelve apostles of the Lamb. Christ chose twelve, one fell, and was replaced by Matthias. These are without a doubt "the apostles", and, when referring to the Gospels, this is often what is directly referred to.

However, Paul was also an apostle. This shows that the title was not limited to the twelve. Biblically, it could mean something similar to "missionary" today.

In fact, by some readings of the text, upward of 25 peopke could be being referred to as apostles, "sent ones", by the Bible itself (see this article here).

So, are there 12, 25, or 70 apostles? The answer is a resounding, Yes! It just depends on what context you are asking the question...

There are three main views on what an apostle is/was as far as I know:

Catholicism
Jesus ordained 12 apostles, who then had the power to choose any number of successors to take on the title of "apostle". They passed on their apostleship to others, who then passed on the apostleship to others, etc. until the today, when the current Roman Catholic clergymen are the apostles.

Some(?) Protestants
Jesus ordained 12 apostles, who were the only ones to ever hold the title. People today can carry out Christ's work as the apostles did through the power of the Holy Spirit.

Mormonism
The Quorum of Twelve Apostles is the second highest governing body in the Church. The Quorums of the Seventy (70 disciples) are auxiliary authorities to the Presidency and the Twelve. “An 'apostle' is an ordained leader in the Melchizedek Priesthood in The Church of Jesus Christ of Latter-day Saints. Apostles are chosen through inspiration by the President of the Church (Prophet), sustained by the general membership of the Church, and ordained by the First Presidency and Quorum of the Twelve Apostles by the laying on of hands. . . . In addition to serving as witnesses of Jesus Christ to all the world (D&C 107:23), as Jesus' apostles did, members of the current Quorum of the Twelve Apostles hold the keys of the priesthood--that is, the rights of presidency (D&C 107:35; cf. 124:128)” (Encyclopedia of Mormonism [1992], 1:59-60, see also–Bible Dictionary: Apostle).

Disciples vs. Apostles
The term "disciple" is generally used for any follower of Jesus who is doing his work.

We can discuss to the world's end what is included in the title apostle.

However, I'd like to point out that there is a special distinction given to the 12 apostles. This is shown in Revelation, where it is revealed that the twelve foundations of the new city has the names of the twelve apostles:

Now the wall of the city had twelve foundations, and on them were the names of the twelve apostles of the Lamb.

~ Revelation 20:14 NKJV

Interesting right? I do wonder if this includes Paul, or is it Matthias (Judas' replacement), since Paul did so much for the spread of the gospel and his epistles make up so much of the NT.

It probably depends on what the specific church calls an apostle, but at least we know that the "seventy disciples" are a distinct group of the twelve (apostles).

From Luke 10:

And after these things, the Lord did appoint also other seventy, and sent them by twos before his face, to every city and place whither he himself was about to come, then said he unto them, `The harvest indeed [is] abundant, but the workmen few; beseech ye then the Lord of the harvest, that He may put forth workmen to His harvest.

The twelve were mostly with him, as far as I remember he was alone with them at the last supper. This shows they were, somehow, special - distinct from the seventy.

LDS view:

In the Church of Jesus Christ of Latter-Day Saints, we have a quorum of twelve apostles and quorums of seventies that assist them. I would (We) see the purpose of the original seventy as similar: Assisting the twelve, going out and preaching. Because really, twelve people is not many.

Neither "apostle" nor "disciple" is an exclusive term: both are descriptive, and you will easily find examples in the New Testament of people from outside Jesus's inner circle described that way.

On the other hand, "the twelve" is clearly a group of twelve individuals (with one late addition replacing Judas Iscariot).

It has become customary to refer to that group as the Apostles (in English, we often use an initial capital like this, but this wasn't a feature of the source languages). So how many Apostles? Christianity down the ages will say 12. How many have been apostles of the Church, or of a church figure, or of Jesus Christ? Many, including those 70 who were sent.

As far as I know, while some Christian denominations make use of the word as badges of approved authority, no mainstream denomination confuses this with inclusion in the category of the twelve. So the idea of Apostolic authority may be contested, but where used it is not a claim to membership of that group of contemporaries of Jesus in the first century.