Q:

Catch the liar(s)

Alice: "Bob is lying, so is Charlie!"
Bob: "Charlie is lying, so is Alice!"
Charlie: "Alice is lying, so is Bob!"
Who is/are lying?

A:

The answer is:

 Exactly one of them is telling the truth but we can't tell which.
 For example, if Alice is truthful, then Bob is a liar because
 he is wrong about her and right about Charlie. Charlie is right about
 Bob but wrong about Alice. This works no matter who you start with.

The other cases:

 No two can be telling the truth because each is calls the other two liars.
 They cannot all be liars because then they would all be telling the truth. 

A:

There are only eight cases so it's easy to check by hand which cases are true:
case 1: They are all lying
This is impossible because that would make their statements true
case 2: They are all telling the truth
this is also impossible because it contradicts all statements
case 3,4,5: two people are telling the truth
This is not possible because person 1 says person 2 lies and vice versa
case 6,7,8: two people are lying
This is possible! The two person that lie, lie about the fact that the truth teller is telling the truth, while the truth teller correctly points to the other two as liars.
So there is one truth teller and two liars, and the truth teller could be anyone

A:

The statement of the puzzle has been changed several times.
My answer below fits with the following version of the puzzle.

Alice: "If Bob is lying, Charlie is lying!"
  Bob: "If Charlie is lying, Alice is lying!"
  Charlie: "If Alice is lying, Bob is lying!"  
Who is/are lying?

We use the words "predecessor" and "successor" with respect to the cycle Alice -> Bob -> Charlie -> Alice.

If person $X$ is lying, then the successor of $X$ is saying the truth (as his/her statement ends with "then $X$ is lying").
If two (or more) persons were lying, then one of them would be the successor of the other one; contradiction to the above statement.
If exactly one of them is lying (say Alice) and the other two (say Bob and Charlie) are saying the truth, we get the contradiction from Charlie: "If Alice is lying, Bob is lying!" (Charlie says the truth; Alice lies; hence Bob must lie).

The only remaining possibility is:  

All three are saying the truth (which indeed is fine).

