The record, released overnight, finds the Queen of Pop singing in Portuguese and exploring the Lisbon-centric genre of fado as well as other contemporary pop sounds.
Much praise centred on the record’s diverse musical palette, with some calling it her best record since 2005’s Confessions On A Dance Floor.
However, some critics found Madonna’s busy approach to song-writing lent the album, her 14th, a disjointed feel.
The Financial Times gave Madame X three stars out of five, saying that 'although it has a scattershot quality, the scattering is done with a devil-may-care bravado'.
Variety magazine said the album sees Madonna giving in to 'the diva within', adding: 'That’s what makes Madame X Madonna’s best album since Confessions On A Dance Floor. She’s confessing again, but this time, she’s not interested in editing herself for mass consumption.'
The Independent gave the album three stars out of five, describing it as the 60-year-old’s 'most exciting, hostile, and, well, bonkers record in ages'.
Rolling Stone magazine also rated the album three stars out of five.
The US publication said Madame X contains a clutch of well-written songs with a distinctly Latin American flavour but to reach them listeners were forced to 'endure' a number of 'disasters'.
Last week, in a flurry of pre-release reviews, the Guardian and the Times gave Madame X four stars out of five, while the Telegraph gave it three.
Fans were broadly happy, with many dubbing it her best work since 2005.
Madonna has conducted a notably large-scale publicity campaign in the lead-up to Madame X’s release.
This came to a head with a two-song performance at this year’s Eurovision Song Contest in Tel Aviv, Israel, last month despite calls for her to boycott the event.
Last month she announced a world tour including a string of shows at the London Palladium.
From PA Entertainment