Chebychev's bias, in its classical form, is the preponderance in ``most'' intervals [2,x] of primes that are 3 modulo 4 over primes that are 1 modulo 4. Recently many generalizations and variations on this phenomenon have been explored. We will highlight the role played by some wide open conjectures on L-functions in the study of Chebychev's bias. Our focus will be on analogues of Chebychev's question to elliptic curves. In the case where the base field is a function field (of a curve over a finite field) we will report on joint work with Cha and Fiorilli and explain how unconditional results can be obtained