The author analyzes certain questions about determining the responsibility of states under the Convention on the Prevention and Punishment of the Crime of Genocide and the latest cases that are being considered before the International Court. Special attention is paid to the decisions of the Court in the cases of Gambia v. Myanmar and Ukraine v. Russian Federation, and to the interpretation of the provisions of the Genocide Convention related to the establishment of the jurisdiction of the Court. In this context, questions of a preliminary nature, such as the existence of a dispute between the parties, the jurisdiction of the Court ratione personae, erga omnes obligations arising from the Convention, the jus standi of the parties before the Court, the relationship between Articles VIII and IX of the Convention, as well as the issue of the use of force to prevent or punishing genocide. In the author's opinion, despite the fact that the analyzed cases have not yet been decided on the merits, the decisions made so far will have a significant impact on the correct and effective application of the Convention on Genocide in the future, in terms of clarifying the prerequisites for establishing the jurisdiction of the Court, as well as preserving the fundamental principles of international rights and rules on the responsibility of states.