This lecture deals with various recent developments concerning the old and very classical concept of topological degree for continuous maps from the circle into itself (also called winding number or index). I will first explain how it can be extended beyond the class of continuous maps. This led to the "accidental" discovery of a simple, but intriguing formula 1 connecting the degree of a map to its Fourier coefficients. The relation is easily justi ed when the map is smooth. However, the situation turns out to be extremely delicate if one assumes only continuity, or even Holder continuity. This "marriage" is more difficult than expected and there are many diffculties in this couple such as the following question I raised: " Can you hear the degree of a map from the circle into itself?" I will also present new estimates for the degree leading to the question : "How much energy do you need to produce a map of given degree?". Many simple looking problems remain open. The initial motivation for this research came from the analysis of the Ginzburg-Landau model in Physics.