Police said the wounded included two foreign tourists, a police officer and the fugitive, Nikos Maziotis. The 43-year-old has been on the run along with his wife Panagiota Roupa since June 2012 following their release from jail in 2011 after serving the maximum 18 months in pre-trial detention.
Maziotis and Roupa were convicted in absentia last year and sentenced to 25 years for participation in Revolutionary Struggle, a group active between 2003 and 2009 and best known for firing a rocket-propelled grenade into the US Embassy and bombing the Athens Stock Exchange. Neither of those attacks caused injuries.
It wasn't immediately clear how today's shootout began in the crowded Monastiraki district, near the city's main Syntagma Square and historic Plaka district.
Local media reported that police were alerted by a store employee about a suspicious customer who appeared to be carrying a gun. The customer, who turned out to be Maziotis, allegedly opened fire and tried to make a getaway in a taxi, but was stopped a short distance away by police and again opened fire.
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Photographs from the scene of the shooting showed the suspect lying in a pool of blood on a sidewalk, his hands handcuffed behind his back, before he was taken in a police-escorted ambulance to a nearby hospital.
Police said the others wounded were a police officer and two bystanders, both foreign tourists. Witnesses at the scene said that an Australian tourist was lightly wounded in the leg. The other tourist was a German who was also lightly wounded, authorities said.
"There wasn't much panic. Store owners and police were telling people to come indoors. But not everyone realised what was going on.