This book chapter is published open access.
The Mordell Conjecture states that a smooth projective curve of genus at least defined over number field admits only finitely many -rational points. It was proved by Faltings in the 1980s and again using a different strategy by Vojta. Despite there being two different proofs of the Mordell Conjecture, many important questions regarding the set of -rational points remain open. This survey concerns recent developments towards upper bounds on the number of rational points in connection with a question of Mazur.