Ukraine insists the United Nations’ highest court has jurisdiction to hear a case alleging that Moscow abused the genocide convention to justify launching its invasion last year.
Kyiv wants judges at the International Court of Justice (ICJ) to order Russia to halt its attacks and pay reparations.
But it appears unlikely Moscow would comply.
Russia has flouted a binding interim order issued by the court last March to end its invasion.
“Russia’s defiance is also an attack on this court’s authority. Every missile that Russia fires at our cities, it fires in defiance of this court,” the leader of Ukraine’s legal team, Anton Korynevych, told the 16-judge panel on Tuesday.
Kyiv filed the case shortly after Russia invaded Ukraine.
“Russia is waging war against my country in the name of this terrible lie that Ukraine is committing genocide against its own people,” Mr Korynevych said.
“This lie is Russia’s pretext for aggression and conquest. Russia has presented no credible evidence. It cannot. In reality, Russia has turned the Genocide Convention on its head.”
Russia outlined its objections to the case on Monday, with the leader of Moscow’s legal team, Gennady Kuzmin, calling it “hopelessly flawed and at odds with the longstanding jurisprudence of this court”.
Ukraine’s case is based on the 1948 Genocide Convention, which both Kyiv and Moscow have ratified.
The convention includes a provision that nations which have a dispute based on its provisions can take that dispute to the World Court.
Russia denies that there is a dispute, a position Ukraine rejects.
In March, the ICC issued a war crimes arrest warrant for Russian President Vladimir Putin, accusing him of responsibility for the abduction of Ukrainian children.
In an unprecedented show of international solidarity, 32 of Ukraine’s allies will make statements on Wednesday in support of Kyiv’s legal arguments.
The court’s panel of international judges will likely take weeks or months to reach a decision on whether or not the case can proceed.
If it does, a final ruling is likely years away.
In his opening statement, Mr Korynevych outlined what is at stake for his country, telling judges that “573 days ago, Russia launched a brutal, full-scale military assault on Ukraine.
“This is a war of annihilation. Russia denies the very existence of the Ukrainian people. And wants to wipe us off the map.”