Judges at the Hague-based International Court of Justice will hand down a ruling at 1300 GMT on Ukraine's request for emergency measures, which it says will help bring stability to its volatile east.
Kiev is also calling on the court to order Moscow to halt what it calls "racial discrimination" against minority groups in the Russian-occupied Crimea peninsula, particularly against its Tatar population.
More than 10,000 people have died in the conflict which entered its fourth year this month, following the ouster of a Kremlin-backed regime in Kiev in February 2014.
Ukraine in January dragged its former Soviet-era neighbour before the ICJ, which was set up in 1945 to settle disputes between countries in line with international law.
Kiev accused Russia of violating the Terrorism Financing Convention and an international treaty against racial discrimination. Moscow rejects the allegations.
In its filing, Ukraine charged Russia with "sponsoring terrorism" by financing pro-Russian separatists and failing to stop military aid from seeping across the border into eastern Ukraine's Donbas region.
It called on the court's 15 judges to rule that "the Russian Federation bears international responsibility" for "acts of terrorism committed by its proxies in Ukraine".
These include the shelling and bombing of civilians and the downing of Malaysia Airlines flight MH17, shot down by a Russian-made Buk-missile over eastern Ukraine on July 17, 2014.
Ukraine wants Russia to pay compensation to all civilians caught up in the conflict -- one of Europe's bloodiest since the 1990s Balkans wars -- as well as to the families of the 298 victims of MH17.
As it can take months for the ICJ to even decide to hear a case, Ukraine also filed an application seeking interim protection measures.
It also urges the tribunal to order Moscow to control its borders with eastern Ukraine and halt racial discrimination in Crimea -- particularly against Tatars -- which Russia annexed in March 2014.
Moscow has strongly denied Kiev's terrorism claims, saying they were "neither factual nor legal" and argued that the ICJ does not have jurisdiction over the case.
The conflict has pushed ties between Moscow and the West to their lowest point since the Cold War.
Georgia filed a similar case against Russia in 2008 after a brief war over two Moscow-backed regions that broke away from Tbilisi's control in the early 1990s.
In an initial ruling, the ICJ ordered both Russia and Georgia to "refrain from any acts of racial discrimination" against ethnic groups.
The court in 2011 declined to hear Georgia's main case, saying it had no jurisdiction. It said Tiblisi should have tried harder to negotiate with its neighbour before seeking the court's intervention.
Olivier Ribbelink, senior researcher at the respected Hague-based Asser Institute told AFP: "You never know what conclusion the court's judges will reach."
He added: "We have been surprised before."
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