Logic, Language, and the Liar Paradox

The Liar paradox arises when we consider a sentence that says of itself that it is not true. If such self-referential sentences exist – and examples like "This sentence is not true" certainly suggest this –, then our logic and standard notion of truth allow to infer a contradiction: The Liar sentence is true and not true. What has gone wrong? Must we revise our notion of truth and our logic? Or ca...