In this talk, we will discuss a beautiful and seemingly elementary conjecture for elliptic curves over the rational numbers discovered and formulated by Sato and Tate in the 1960s. We will start from the famous Hasse-Weil bound (Deligne’s purity theorem) and Serre’s L-function argument reduction of the conjecture to some Langlands philosophy in terms of complex/harmonic analysis. Then I will sketch the proof, which is of similar flavour to the solution to Fermat's last theorem. This was completed and generalized to holomorphic modular forms of higher weights (e.g. the Ramanujan tau function) in a series of papers. If time permits, I will discuss the recent solution of a generalized conjecture for elliptic curves over imaginary quadratic fields by 10 mathematicians and explain some different geometry behind this generalization.