  In this paper we discuss the generalizations of the concept of Chebyshev's
bias from two perspectives. First we give a general framework for the study of
prime number races and Chebyshev's bias attached to general $L$-functions
satisfying natural analytic hypotheses. This extends the cases previously
considered by several authors and involving, among others, Dirichlet
$L$-functions and Hasse--Weil $L$-functions of elliptic curves over
$\mathbf{Q}$. This also apply to new Chebyshev's bias phenomena that were
beyond the reach of the previously known cases. In addition we weaken the
required hypotheses such as GRH or linear independence properties of zeros of
$L$-functions. In particular we establish the existence of the logarithmic
density of the set $\lbrace x\geq 2 : \sum_{p\leq x} \lambda_{f}(p) \geq 0
\rbrace$ for coefficients $(\lambda_{f}(p))$ of general $L$-functions
conditionally on a much weaker hypothesis than was previously known.
