The Toffees sacked Frank Lampard last week with the club 19th in the Premier League and two points from safety.
Dyche, 51, had been out of work since he was dismissed by Burnley last April after almost 10 years at the Clarets.
He is Everton majority owner Farhad Moshiri’s seventh managerial appointment since buying into the club in February 2016.
Dyche guided Burnley to two promotions from the Championship and two top-half Premier League finishes, including a first European qualification in 51 years, after they finished seventh in 2017-18.
His immediate task will be to guide Everton out of the relegation zone and to safety with 18 games remaining.
Everton are next in action on 4 February, when they host Premier League leaders Arsenal at Goodison Park, with a game against Merseyside rivals Liverpool on 13 February.
Former Leeds United manager Marcelo Bielsa was also a contender for the job and held talks with Everton’s hierarchy.
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