The Hitchin fibration for algebraic curves has been object of intensive studies from which arised important applications ranging from geometry to number theory. The Hitchinfibration for higher dimensional varieties is much less studied. In this talk, I will give an overview of my joint work with T.H. Chen on the geometry of the Hitchin fibration for algebraic surfaces. In the first hour, I will recall the basic features of the one-dimensional case which on wish to generalize to higher dimension. In the second hour, I will present the little we know about the higher dimensional case with emphasis on the case of surfaces and formulate questions.