In the capital Dakar, security forces fired tear gas to disperse the crowds.
Current opposition leader Khalifa Sall, who is not related to the president, earlier called the election delay a “constitutional coup”.
Opposition called from state-wide protests. However, Khalifa Sall denounced the violence between protesters and police. “The hearts of all democrats bleed at this outburst of clashes provoked by the unjustified halting of the electoral process,” he said.
Earlier president Sall said he was ready to step down, but wanted to leave the country stable and peaceful.
Macky Sall has said he is not planning to run for office again – but his critics accuse him of either trying to cling on to power or unfairly influencing whoever succeeds him.
The Economic Community of West African States, ECOWAS has over the past days condemned the postponement of Presidential election in Senegal, saying the development would further destabilize the region.
Senegal is one of the few countries in Africa that never witnessed a military coup.