Are Alice, Bob and Charlie telling the truth, or lying?

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Alice, Bob, and Charlie make the following statements:

Alice: Bob is lying.
Bob: Charlie is lying.
Charlie: 1 + 1 = 2.

Who is telling the truth? Who is lying? Explain your answer.

Now Alice, Bob, and Charlie make the following statements:

Alice: Bob is telling the truth.
Bob: Alice is telling the truth.
Charlie: Alice is lying.

What are the possible numbers of people telling the truth? Explain your answer.

They now make the following statements:

Alice: Bob and Charlie are both lying.
Bob: Alice is telling the truth or Charlie is lying (or both).
Charlie: Alice and Bob are both telling the truth.

Who is telling the truth and who is lying on this occasion? Explain your answer.