The Frenchman underwent a liver transplant last year but did not have his contract renewed at Barcelona.
Abidal joins an impressive list of signings this summer for Monaco, which includes James Rodriguez, Radamel Falcao and Joao Moutinho.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.