A Lie algebra is called finitary if it consists of finite-rank linear transformations of a vector space. The authors classify all infinite-dimensional finitary simple Lie algebras over an algebraically closed field of characteristic not 2 or 3. They also do the same for finitary irreducible Lie algebras. The simple algebras of the classification are finitary special linear algebras, finitary orthogonal algebras and finitary symplectic algebras. Finitary irreducible algebras, in addition to the simple algebras, also include finitary general linear algebras.