Bemba lost presidential elections to Kabila in 2006 and was later accused of treason when his bodyguards clashed with the army in Kinshasa.
In 2007 he fled to Belgium, where he had spent part of his youth.
He was then arrested in Europe on an ICC warrant for war crimes committed by his private army in the Central African Republic from 2002-3, when its then-president Ange-Felix Patasse sought his help to repel a coup attempt.
Bemba, who became vice-president of an interim government from 2003 to 2006, was born on November 4, 1962 at Bogada in the northwest Equateur province.
His father was a rich businessman close to dictator Mobutu Sese Seko, who ruled from 1965 until he was ousted in 1997.
He left the country in 1997 after Laurent-Désiré ousted Mobutu. A war followed a year later. It lasted till 2003 and drew foreign support on rival sides.
Bemba became leader of the Congolese Liberation Movement (MLC) rebels, a 1,500-strong force backed by neighbouring Uganda and opposed to the Kabila regime.
The MLC, now a political party, on Tuesday urged the ICC to be more "precise" on its ruling, saying Bemba had been unfairly banned by the Constitutional Court. It also called an emergency meeting to review the situation.
"The final word does not rest with the Constitutional Court but with the Congolese people," said Jean-Jacques Ntula, a resident of Mbandaka, the main city in Equateur province, a Bemba stronghold.
After the Congolese war ended, Bemba laid down his arms and was awarded one of four vice-presidential posts shared out among war-time rivals in a transitional government.