Simple Lie algebras over the complex numbers were classified by Killing and Cartan in the late 1800s. Classifying the simple ones over an algebraically closed field of prime characteristic turned out to be much harder, with the classification in characteristic ¿ 7 only conjectured in 1966 and finally solved in the late 1980s. Characteristics 5 and 7 have only been settled in the last few years. We survey these results together with a recent result on E8 in characteristic 5.