The 47-year-old had been out of work since being sacked by PSG in December.
Chelsea confirmed on their official website: "Thomas Tuchel has been appointed as the new Chelsea head coach."
Tuchel said: "I would like to thank Chelsea FC for their confidence in me and my staff.
"We all have the greatest respect for Frank Lampard's work and the legacy he created at Chelsea. At the same time, I cannot wait to meet my new team and compete in the most exciting league in football. I am grateful to now be part of the Chelsea family - it feels amazing!"
Despite leading PSG to a first Champions League final, where they were defeated 1-0 by Bayern Munich in August, Tuchel was replaced by Mauricio Pochettino after overseeing four Ligue 1 defeats before the mid-season break. PSG were third, a point behind Lyon and Lille, when Tuchel was ousted.
The German won two Ligue 1 titles, the Trophee des Champions twice, the Coupe de France and the Coupe de la Ligue during his two and a half years at the helm in Paris.
Prior to that he spent two years with Dortmund, winning the DFB-Pokal in 2016-17 but failing to dethrone Bayern in the Bundesliga.
He left both his previous positions following a breakdown in relations with senior leadership figures.
Tuchel had a disagreement with Dortmund CEO Hans-Joachim Watzke after a Champions League game against Monaco was pushed back by just one day after the Bundesliga side's bus was attacked.
At PSG he was involved in a public dispute with sporting director Leonardo over the club's transfer policy and reportedly claimed he felt more like "a politician rather than a coach", though he later said that remark was mistranslated.
Chelsea will be keen for Tuchel to make a quick impact on the pitch, with the Blues having let Lampard go after a run of two wins in eight Premier League matches.
They face Wolves at Stamford Bridge on Wednesday before entertaining Burnley in the top flight on Sunday.