The Anthony's chain of food outlets included restaurants in the city's Corn Exchange building and a patisserie in the Victoria Quarter.
The firm was founded in 2004 by chef Anthony Flinn and his father, also called Anthony Flinn.
Lines Henry Ltd have been appointed as administrators and said there was not enough money for the firm to continue trading.
Michael Simister, of Lines Henry, said: "It is sad when such a well-known business runs into difficulties.
"Unfortunately, the business has had to close, with the loss of 70 jobs, because there is not enough money available to continue trading."
He added despite good reviews the business had struggled with increased competition and difficult economic conditions.

West Midlands Police said the man had been detained in Reading and the woman in Wokingham by counter-terrorism officers at about 19:30 BST.
The pair, believed to be friends, were taken to a police station in the West Midlands for questioning.
The force said the arrests had been intelligence-led.
The suspects did not pose any imminent threat, a spokesperson added.
Supt Robin Rickard from Thames Valley Police said the arrests "were part of an ongoing investigation by West Midlands Counter Terrorism Unit and were pre-planned".
"Officers from Reading and Wokingham are providing support and there are searches taking place at the suspects' home addresses," he said.
The failure of a restaurant business in Leeds has led to the loss of 70 jobs.
[NEXT_CONCEPT]
A 24-year-old man and a 23-year-old woman have been arrested in Berkshire on suspicion of preparing for acts of terrorism in the UK.