The Mordell--Weil Theorem states that the rational points of an abelian variety are finitely generated. The proof is in two steps: first, we will reduce the problem to the weak Mordell--Weil Theorem using the theory of heights. Then we will prove the weak Mordell--Weil Theorem from the finiteness of the Selmer group. The talk will be accessible to anyone who has seen some algebraic number theory.