Bosnia says Belgrade was responsible for crimes of genocide on its territory during the early 1990s Bosnian war.
Belgrade denies its intention was to wipe out Muslims in eastern Bosnia.
The EU is also exerting pressure on Serbia, as foreign ministers threaten to freeze association talks unless it co-operates over war crimes suspects.
The ministers did not set a specific deadline, but indicated that they wanted fugitive suspect Ratko Mladic handed over by the end of March. He is accused of genocide and other crimes committed during the Bosnia war.
The ministers warned that negotiations with Serbia scheduled for April could be postponed if the former Bosnian Serb general was not surrendered to the UN war crimes tribunal.
Serbia's Deputy Prime Minister Miroljub Labus said the security services had been told to arrest Gen Mladic and he hoped this would happen by the end of March.
The genocide case against Serbia and Montenegro is being heard at the International Court of Justice (ICJ), also in The Hague.
On Monday, hundreds of survivors of the war held a vigil outside the court and read out the names of Bosnian Muslims killed by Serb forces.
The hearings at the ICJ or World Court, which mediates in disputes between states, are scheduled to run until 9 May, but a ruling is not expected until the end of the year.
The BBC's Geraldine Coughlan says if Bosnia wins the genocide case, it will seek compensation from Serbia, which could run into billions of dollars.
Phon van den Biesen, one of the lawyers acting for Bosnia, said: "They really destroyed important parts of each and every town which would be relevant for a comeback of the non-Serb population.
"So the destruction which has been brought about after the actual takeovers of cities and towns is enormous."
Serbia will deny that the state - rather than a group of individuals - had the specific intent to wipe out the Muslim population of eastern Bosnia.
Bosnia's case will focus on the 1995 Srebrenica massacre, already established as genocide by the International War Crimes Tribunal for the Former Yugoslavia.
Proving the Serbian nation's responsibility for the most serious war crime of genocide is an historic challenge for the Bosnian legal team, says our correspondent.
The hearings have been delayed for over a decade, since Belgrade filed a series of counter-claims and disputed the court's authority.