Following a suggestion by I. M. Gelfand, and motivated by questions arising from the Ginzburg-Landau model - I discovered a few years ago, an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth.  However, the situation turns out to be extremely delicate if one assumes only continuity, or even Holder continuity. I will present recent developments  and open problems.  I will also discuss new estimates for the degree leading to unusual characterizations of Sobolev spaces.  This also raises challenging questions connected to Gamma-convergence.