Šťastný started his career with the Quebec Nordiques in 1980 after defecting from communist Czechoslovakia along with his brother Anton Šťastný, who also joined the Nordiques; they were joined by their younger brother Marián the next season. He was a prolific scorer, scoring 450 goals and adding 789 assists for 1239 points, becoming the first European-born and trained player to reach the 1000 points plateau. Stan Mikita, who was born in Slovakia too, but trained in Canada.
Peter comes from a true hockey family. His two younger brothers played with him in the NHL, and his older one, Vladimír, was assistant coach for the Slovak National Team. His sons Paul and Yan currently play in the NHL; Paul finished as a finalist for the Calder Memorial Trophy in 2006-07, 16 years after his father won it.