Are all abelian groups free?

Could you please clarify for me if it is accurate to state that all abelian groups are inherently free? It's a question that has been lingering in my mind, as I understand that abelian groups possess a certain level of commutativity, but I'm unsure if this automatically translates to them being free groups. Could you elaborate on the relationship between these two concepts and whether there are any exceptions to this potential rule?