A product of the acclaimed Santa Clara (California) Swim Club, Schollander was noted for his speed, perfection of stroke, and effortless crawl. He broke several U.S. and world freestyle records in the early 1960s and in 1963 became the first to swim 200 metres in less than 2 minutes, improving the time for this distance in nine record-breaking swims between July 1963 and August 1968. In 1964, prior to the Tokyo Olympics, he lowered the world 200-metre freestyle standard to 1 min 57.6 sec, and he also broke the world 400-metre freestyle record before reducing it once more at the Olympics.

At the Tokyo Games, Schollander set a world record (4 min 12.2 sec) in the 400-metre freestyle and an Olympic record (53.4 sec) in the 100-metre freestyle event and was a member of the U.S. teams that established world records while winning the 4 × 100-metre and 4 × 200-metre relay races. At the 1968 Olympic Games in Mexico City, he won a silver medal in the 200-metre freestyle, after setting a world record of 1 min 54.3 sec in the pre-Olympic U.S. team trials. He also was a gold medal winner in Mexico City with the 4 × 200-metre relay team. In 1983 Schollander was among the first group of athletes to be inducted into the U.S. Olympic Hall of Fame.