The notion of a normal cone of a given set is paramount in optimization and variational analysis. In this work, we give a definition of a multiobjective normal cone which is suitable for studying optimality conditions and constraint qualifications for multiobjective optimization problems. A detailed study of the properties of the multiobjective normal cone is conducted. With this tool, we were able to characterize weak and strong Karush-Kuhn-Tucker conditions by means of a Guignard-type constraint qualification. Furthermore, the computation of the multiobjective normal under the error bound property is provided. The important statements are illustrated by examples.