Everton hasn't won any silverware since 1995 but, having won nine top-flight league titles, five FA Cups and a European trophy, it is regarded as one of England's most famous clubs.
Everton are now two points above the relegation zone going into Wednesday night's match against West Ham.
Former England boss Allardyce ruled himself out of the running earlier this month, claiming he did not feel the Toffees were convinced he was the right man.
However, after a few months out of the game and after much speculation, Allardyce clearly had a change of heart as he was finally announced as the new Everton boss on Thursday.
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A salary close to the £6 million per season that Koeman earned is understood to have been agreed as the 63-year-old prepares to take the reins at Goodison Park.
Everton are now sitting 17th in the table, two points ahead of the Hammers.
The club's prospects had appeared so different at the beginning of the season with Everton having invested heavily following last season's impressive seventh-place finish. However, the club started poorly in both the Premier League and the Europa League. But, after taking four points from their first two games, Everton managed the same tally from their next seven fixtures and Koeman was sacked. So with that in mind it's no wonder Everton are throwing a lot of money at a manager who's never been relegated.
Everton's communications department were unaware of this and an embarrassing situation was made worse when bookmaker Paddy Power released quotes given to them by Allardyce two weeks ago in which he dissected the team's problems. But that turned sour after only one match, with Allardyce stepping down in September 2016 following a newspaper sting.