The theory of Lie algebras can be categorified starting from a new notion of `2-vector space', which we define as an internal category in the category of vector spaces. We define a `semistrict Lie 2-algebra' to be a 2-vector space L equipped with a skew-symmetric bilinear functor [, ]: L x L -> L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the `Jacobiator', which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang-Baxter equation.