The Mumford-Tate conjecture is a deep conjecture which relates the arithmetic and the geometry of  abelian varieties defined over number fields. The results of Moonen and Zarhin indicate that this conjecture holds for almost all absolutely simple abelian fourfolds. The only exception is when the abelian varieties have no nontrivial endomorphism. In this talk we will begin with an introduction to the Mumford-Tate conjecture and a brief summary of known results towards it. Then we sketch a proof of this conjecture in the above 'missing' case for abelian fourfolds.