In each instance, the suspect carried what is believed to be a handgun and made demands for cash.
The man, who is described as being around six ft and of medium build, was wearing a balaclava and gloves.
Detectives are currently linking five separate incidents, four of which happened on Tuesday in the Bramley and Hough End areas.
The first took place at the Swinnow Shopping Centre just after 7pm, closely followed by an incident at a nearby newsagents on Morris Lane at 7.17pm.
A further offence took place at 9.22pm at a bookmakers on Pudsey Road, which was followed by another robbery at a post office at Wyther Park.
A man also escaped with a quantity of cash during an armed robbery which happened on Hyde Park Road in the Burley area at 2pm yesterday.
Detective Inspector James Entwistle, of Leeds District CID, said: “We have been making extensive CCTV enquires in the areas where each of these offences have occurred, which is helping us to trace the individual responsible.
“Today we have released CCTV images retrieved from that footage to assist in tracing the suspect, who we believe has been involved in each of these armed robberies.
“In particular, we have shared the audio from one the incidents, as the voice captured is quite distinctive.
“Someone may be able to recognise this and enable us to quickly identify the individual behind this series of crimes."
He said he understood why people in the local area may be worried about the spate of offences, but his team was doing all it could to find the person responsible.
“We will not accept criminal activity on the streets of Leeds and officers will be providing high visibility reassurance to residents and local businesses," he said.
“If you have any information or have witnessed one of these offences, please contact the non-emergency number 101 or Crimestoppers in confidence on 0800 555 111.
“Likewise, if you can tell us anything emerging from the CCTV footage, please get in touch with us.”
The force also has a live chat facility which is accessible via the Contact Us section of the force website.