The Netherlands-based International Court of Justice (ICJ), will begin hearings on Monday in a case that is expected to define the nature of the 1992-95 Bosnian conflict, which claimed tens of thousands of lives.
At the centre of the case are claims by many Bosnians that the leaders of a crumbling Yugoslavia controlled and financed Serb forces during the war in an attempt to maintain a grip on the territory.
Belgrade denies it sent its own troops or backed Bosnian Serb forces and describes the conflict as a civil war fought between Bosnia's Orthodox Serbs, Muslims and Catholic Croats.
The case that Sarajevo brought before the UN's highest court in 1993 - at the peak of the war - directly accuses Belgrade of recruiting, training and equipping Serb fighters who murdered, raped and tortured civilians during the Bosnian war.
The hearings are another blow to Belgrade at a time when it already faces the threat of renewed international isolation over its failure to capture war crimes suspect Ratko Mladic, a former Bosnian Serb general believed to be hiding in Serbia.
Search for truth
Friday's demonstrations were organised by a number of associations of war victims and veterans in predominantly Muslim cities.
Irfan Ajanovic, the head of the association of survivors of wartime detention camps run by Bosnian Serbs, said: "We expect the truth finally to be established.
More than 8000 men and boys
were slaughtered in Srebrenica
"We are not aiming to have one nation be declared responsible but we want a certain kind of politics to be condemned, genocide and crime to be condemned ... so that this never happens to anyone in the world again."
Amela Selmanovic, one of those who took part in Friday's demonstrations, said: "We only want a court to say what happened here.
"We are not afraid of the truth. Only those who are afraid of the truth are running away from it."
The hearings before the UN court were delayed for more than a decade by a series of counter claims filed by Belgrade.
If the court decides in Bosnia's favour, Serbia-Montenegro could be ordered to pay billions of dollars in compensation.
On Friday the government in Belgrade had no comment on the case.
There is opposition to the case in Bosnia as well, from Bosnian Serbs who have made several attempts to block it.
Bosnian Serb leaders succeeded in stopping state funds from being used to finance the case.
"We are not afraid of the truth. Only those who are afraid of the truth are running away from it"
The 1995 peace agreement that ended Bosnia's war sealed the country's wartime ethnic divisions, splitting it into two mini-states - one for Serbs and the other shared by the country's Muslims and Croats.
Dragan Cavic, the president of Bosnia's Serb Republic, said on Friday the country's lawsuit against its neighbour is "absurd".
He said it also suggests that the Bosnian Serb mini-state was the result of aggression and genocide, and in so doing, threatens its legitimacy.
The Bosnian side will largely base its case on findings of the UN war crimes tribunal for the former Yugoslavia, where prosecutors in the trial of Slobodan Milosevic, the former Yugoslav president, have sought to link Belgrade to atrocities in Bosnia and elsewhere in the Balkans.
The slaughter of as many as 8000 Muslim men and boys in the Bosnian enclave of Srebrenica will likely be central to their arguments that Belgrade was deeply involved in the country's bloodshed.
Those allegations were further fuelled last year when a video emerged showing six civilians being shot to death in execution-style killings that were part of the 1995 Srebrenica massacre.
The hearings at the UN court are scheduled to last until 9 May. A ruling will likely come several months later.
The ICJ is the UN's venue for resolving disputes between nations.