Bronze does not rust. The term "rust" refers strictly to the oxidation of iron and its alloys. Bronze is made primarily from copper and tin, with no iron added.
Though it does not rust, exposed bronze is subject to oxidation. Copper is the major component of bronze, and it passes through several stages of oxidation in the presence of oxygen and an electrolyte such as rain water. Eventually, the copper in bronze forms a green patina on its surface that prevents further degradation of the underlying metal. Bronze also decays in contact with sulfurous compounds and chlorine, such as those present in sea spray.