How many smooth structures are there on a sphere? For dimensions at least 5, Kervaire--Milnor solved this problem in terms of another problem in algebraic topology: The computations of stable homotopy groups of spheres. In this talk, I will discuss recent progress on this problem in algebraic topology and its applications on smooth structures, which includes the following result with Guozhen Wang, building up on Moise, Kervaire--Milnor, Browder, Hill--Hopkins--Ravenel: Among all odd dimensions, the n-sphere has a unique smooth structure if and only if n = 1, 3, 5, 61. I will also discuss some recent progress towards the Kervaire invariant problem in dimension 126.