When Ullman was 8 years old he lived close to an American military base in Edmonton, Alberta, Canada. The base had an ice rink, it was there he first took up ice skating after his father made arrangements for his kid to have a little fun. By 10 years old he had taken up the sport that would make him famous. By the age of 16, Ullman was playing with the Edmonton Oil Kings, who were sponsored by the Detroit Red Wings which meant they owned his rights. He turned pro at the age of 18, then spent a couple seasons with the Red Wings top minor league affiliate, the Edmonton Flyers. By the time he was 20 years old he was ready to take the next step and he joined the Red Wings for the 1955-'56 season.
When Ullman arrived, he joined a team that had dominated the decade by winning 4 of the last 6 Stanley Cup's. The kid was in awe to be playing with such great names like Gordie Howe, Terry Sawchuk, and a variety of other great players. In his first full season he didn't get the ice time that he would later receive, but he still recorded 9 goals with 9 assists and proved he had what it took to play at the highest level. The following year Norm scored 16 times and recorded 36 assists, he became a model of consistency over the next few years and was selected to the All Star game in the 1960 season, that was the first season he finished in the top ten in goals scored. He would crack the top ten three seasons in a row before taking the top spot in the '64-'65 season, with a 42 goal performance. That year he was just 4 points shy of the league's scoring title that went to Stan Mikita. Ullman had become everything a team could want in a player, he scored goals, could take and give a hit, and was recognized regularly by being selected to the All Star team.
In his 20 year NHL career he scored 490 goals, 739 assists, for a grand total of 1,229 points. Despite his consistent play he was not able to add the Stanley Cup to his resume, but he was recognized as one of the best players to play the game when he was inducted into the Hockey Hall of Fame in 1982, Ullman called it the greatest honor of his lifetime.