Due to an application of gauge theory to smooth $4$-manifolds, it has been known that some topological $4$-manifolds do not admit a smooth structure and some smooth $4$-manifolds, for example elliptic surfaces, admit an exotic smooth structure.

In this talk we survey known results on exotic smooth structures of $4$-manifolds. And then we conclude that most known simply connected, closed, irreducible, smooth $4$-manifolds with $b_{2}^{+}$ large enough admit infinitely many distinct exotic smooth structures. Explicitly, for each known simply connected, closed, irreducible, smooth $4$-manifold with $b_{2}^{+}$ large enough, we present a family of infinitely many, both symplectic and non-symplectic, $4$-manifolds which are homeomorphic, but not diffeomorphic, to a given $4$-manifold.