The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is a 'stable range' of indices which one understands better than the unstable part.
I think that there is an analogy (have to be careful with that word) to the distribution of primes: It seems that there exists a general pattern but no one has found it yet. It is a construction producing an infinite list of numbers (or groups) but no numbers were put into it. Such a thing always fascinates me.
The largely unknown prime pattern leads to applications in cryptography for example. Are there similar applications of the knowledge (or not-knowledge) of the homotopy groups of spheres? Are there applications to real natural sciences or does one study the homotopy groups of spheres only for their inherent beauty?