Abstract: From the beginning of the theory in Banach spaces, people have tried to find analogues of orthonormal bases in Hilbert spaces. (Orthonormal bases themselves are, from the point of view of an analyst, natural analogues of bases in vector spaces.) I will present the two most natural concepts of bases in Banach spaces, and I will discuss the question:''Do bases exist for all Banach spaces?'' I will also explain the contribution that W.T.Gowers (jointly with B.Maurey) made in this area, which was that he was able to give a way of constructing (almost) arbitrarily strange and fabulous Banach spaces. In particular, I will explain how Gowers and Maurey answer the fourth of the four most natural questions relating to bases in Banach spaces. If time permits, I will discuss how his construction can be used to solve several other questions dating back to Banach.