1-motives have been introduced by Deligne in the 1970´s as the simplest kind of mixed motives; basically they are the only ones having a simple and concrete geometric description. In recent years there has been a resurgence of interest in them, and they have been successfully applied to arithmetic questions as well, such as generalizations of the classical duality theorems of Poitou and Tate, or finding rational points on algebraic varieties. In these lectures I shall try to explain why 1-motives arise naturally in various contexts, and explain some of the above-mentioned applications.