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.