Lie algebras constitute a linearized framework to study the local properties of Lie groups. In this chapter, the first basic notions about Lie algebras are introduced. In fact, when dealing with differential equations, the algebraic structure of their Lie symmetries (spanning a Lie algebra) plays a crucial role both for the determination of their solutions, their transformation to equivalent differential equations, and their characterization. We introduce the most important objects: Lie brackets, structure constants, Lie subalgebras, ideals, derived algebras, Lie homomorphisms, ..., allowing us to define solvable, nilpotent, simple, and semisimple Lie algebras. Solvable and Abelian Lie subalgebras are particularly relevant in many applications of Lie symmetries of differential equations. Several examples are also given. Moreover, introducing the adjoint representation and the inner automorphisms, it is shown how to construct an equivalence relation between subalgebras leading to the concept of optimal systems of Lie subalgebras; in turn, similar subalgebras yield a classification of Lie subgroups up to conjugacy relation.