The notion of &rdquo / Planar functions&rdquo / goes back to Dembowski and Ostrom, who introduced it in 1968 first time to describe projective planes with special properties in finite geometry. Recently, they attracted an interest from cryptography because of having an optimal resistance to differential cryptanalysis.This thesis is based on the paper &rdquo / New semifields, PN and APN functions&rdquo / by J&uuml / rgen Bierbrauer. The whole purpose of this thesis is to understand and present a detailed description of the results of the paper of Bierbrauer about planar functions. Here and throughout this thesis &rdquo / new&rdquo / means &rdquo / new&rdquo / in the paper of Bierbrauer. In particular we have no new constructions here and we only explain the results of Bierbrauer.