Almost complex structures on spheres

In this paper we review the well-known fact that the only spheres admitting an almost complex structure are S2 and S6. The proof described here uses characteristic classes and the Bott periodicity theorem in topological K-theory. This paper originates from the talk “Almost Complex Structures on Spheres” given by the second author at the MAM1 workshop “(Non)-existence of complex structures on S6”, held in Marburg from March 27th to March 30th, 2017. It is a review paper, and as such no result is intended to be original. We tried to produce a clear, motivated and as much as possible self-contained exposition.