From Wikipedia:

Drifting as a driving technique is documented as early as the 1930s as being used by drivers of the Grand Prix cars of the day. At least one piece of extant period footage used to promote the sale of a rare Auto Union D-Type racer clearly depicts the driver throwing his vehicle into a controlled drift to navigate a bend in the road racing track.

Japanese Adaptation

Modern drifting as a sport started out as a racing technique popular in the All Japan Touring Car Championship races. motorcyclist legend turned driver, Kunimitsu Takahashi, was the foremost creator of drifting techniques in the 1970s. He is noted for hitting the apex (the point where the car is closest to the inside of a turn) at high-speed and then drifting through the corner, preserving a high exit speed.  This earned him several championships and a legion of fans who enjoyed the spectacle of smoking tires.  The bias ply racing tires of the 1960s-1980s lent themselves to driving styles with a high slip angle. As professional racers in Japan drove this way, so did the street racers.

D1 Grand Prix

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Western Adoption

One of the earliest recorded drift events outside Japan was in 1993, held at Willow Springs Raceway in Willow Springs, California hosted by the Japanese drifting magazine and organization Option. Inada, founder of the D1 Grand Prix in Japan, the NHRA Funny Car drag racer Kenji Okazaki and Keiichi Tsuchiya, who also gave demonstrations in a Nissan 180SX that the magazine brought over from Japan, judged the event with Rhys Millen and Bryan Norris being two of the entrants.  Drifting has since exploded into a massively popular form of motorsport in North America, Australasia, and Europe.

Formula Drift

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Present Day

Drifting has evolved into a competitive sport where drivers compete mostly in rear-wheel-drive cars, to earn points from judges based on various factors. At the top levels of competition, the D1 Grand Prix in Japan pioneered the sport. Others such as Formula D in the United States, and the NZ Drift Series in New Zealand have come along to further expand it into a legitimate motor sport worldwide.  The drivers within these series were originally influenced by the pioneers from D1 Japan and are able to keep their cars sliding for extended periods of time, often linking several turns.

NZ Drift

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