The 29-year-old had an indifferent spell on loan at Manchester United last term, scoring four goals in 29 games.
"I am very happy to be joining Chelsea and can't wait to start training and help with our aim of retaining the league title and being successful in Europe," Falcao told the club website.
Falcao, who failed to score for Colombia at this year's Copa America where they reached the quarter-finals, previously played for River Plate, Porto and Atletico Madrid.
He became one of Europe’s most feared strikers, having started his career as a teenager in Colombia before joining Argentine giants River in 2001.
Eight years later, having established himself as one of South America’s deadliest strikers, Falcao joined Porto where he made his Champions League debut against Chelsea.
He soon began scoring regularly, netting 72 goals for Porto in two seasons from 2009 and won successive Europa League titles in 2011 and the following year after joining Atletico.
Falcao hit 70 goals in two seasons for the Spaniards and got a stunning hat-trick against Chelsea in the 2012 European Super Cup.
He left Madrid for Monaco in 2013 for almost $80 million (51 million pounds), enjoying a fine start to the campaign before suffering anterior cruciate ligament damage in January that ruled him out for the rest of the season and the 2014 World Cup.
After two goals in three games at the beginning of last season, he joined United, helping Louis van Gaal’s side finish fourth to qualify for the Champions League.
Falcao will be reunited at Chelsea with former Atletico team mates Thibaut Courtois, Filipe Luis and Diego Costa, the player he played in attack with during their 2013 King's Cup triumph over Real Madrid, plus Colombia colleague Juan Cuadrado.
(Reporting by Ed Osmond and Ken Ferris, editing by Pritha Sarkar)
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