The problem of free will, as classically defined, is that since each event is caused by a chain of antecedent events, nothing we do is of our own free will; if we do have free will, its hard to see how it could come about.
Humankind has been at this problem for millennia. Instead of listing famous arguments, for and against free will (or different nuanced interpretations of this problem) it would suffice to say that we are nowhere near to solving it.
But I cannot resist mentioning my favorite “solution”, by the great American philosopher, William James. James believed in free will and had an amusing argument against those who argue against free will. Paraphrasing and simplifying, James focussed on the futility of an opposing argument. Why argue against free will? It would make no difference to the person you are trying to convince because she has no free will, and the fact that you are arguing against free will, implicitly means that you too believe in free will!
The only response to this is to accept that one is arguing against free will because one is being forced to do it 😛
In line with James argument, I made the following animation to make a similar point in a humorous manner.