Computing and understanding the stable homotopy groups of spheres is a fundamental problem in algebraic topology. It has many connections to other subjects of mathematics. In these four lectures, I will survey on two recent results regarding applications to smooth structures on manifolds. The first result, joint work with G. Wang, states that the spheres in dimensions 1, 3, 5, and 61 are the only odd dimensional ones that admit a unique smooth structure. The second result, joint work with M. Hopkins, J. Lin and X. D. Shi, is a ``10/8 + 4" theorem of the geography problem for simply connected spin 4-manifolds. Both results are proved through computations of stable homotopy groups of spheres. I will talk about part of the techniques used in the proofs if time permits. The audience should be familiar with basic concepts in algebraic and differential topology, but is not required to be familiar with stable homotopy theory.