In my three talks, I will explain the construction of a twisted Donaldson invariant. We also include basic subjects on both gauge theory and non commutative geometry. Below we describe the details of three talks. This is joint work with H. Sasahira and H.Wang. [3/19(Mon) 11:00~12:10 Place: 24-112] Lecture I: Rough description of basic gauge theory We quickly review Yang-Mills gauge theory and Donaldson theory. [3/21(Wed) 11:00~12:10 Place: 24-112] Lecture II: Twisted Donaldson invariant 1 We define a twisted Donaldson’s invariant using the Dirac operator twisted by flat connections when the fundamental group of a four manifold is free abelian. We also verify non triviality of the invariant by presenting some exotic pairs which are obtained from them. [3/22(Thur) 14:00~15:10 Place: 25-103] Lecture III: Twisted Donaldson invariant 2 Using Connes-Moscovici’s index theorem, we introduce the construction of twisted Donaldson’s invariant when fundamental group is non-abelian. Non commutative geometry is a useful tool in the study of topology of Riemannian manifolds. Taking into account of the fundamental group in the formulation of a topological invariant, one can obtain refined topological invariants involving the C^*-algebra of the fundamental group. For example, the Novikov conjecture on homotopy invariance of higher signatures has been developed extensively using non-commutative geometry. We apply the method of non commutative geometry to the construction of twisted Donaldson’s invariant.