Asper was born in Minnedosa, Manitoba, the son of musicians who had emigrated from the Ukraine. In 1957 he received his Bachelor of law degree, and was called to the bar shortly thereafter. He founded the firm of Asper, Freedman & Co. in 1959. In 1970 he wrote The Benson Iceberg: A critical analysis of the White Paper on Tax Reform in Canada.
His media empire started with the Winnipeg television station CKND in 1975. CanWest grew to encompass the Global Television Network, the daily newspaper National Post and over 60 other Canadian newspapers.
Asper was noted for his fierce loyalty to Manitoba, refusing enticements to move east to Toronto. He was the leader of the Manitoba Liberal party from 1970 to 1975. The faculty of management at the University of Manitoba renamed itself the Asper School of Business in 2000. He was also a noted philanthropist, making major donations to the arts and education; in 2001 he donated $5 million (Can) to the St. Boniface Hospital & Research Foundation in Winnipeg. Asper became an Officer of the Order in Canada in 1995.
He was one of Canada's most prominent Jews, and was well-known for his strong faith and support for the State of Israel. He would occasionally pen editorials defending the nation in his various papers. He also advocated against unaccountable public broadcasting media, especially the CBC, which critics have accused of simply attempting to eliminate competition to his business and his political opinions.
Asper was also a close friend of many of Canada's prominent political and business elites, including Jean Chrétien and Paul Martin. He had three children, David, Leonard and Gail, all of whom became lawyers.