  We study hyperelliptic curves arising from Chebyshev polynomials. The aim of
this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic
curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2
= \varphi_{d}(x)$ is maximal over the finite field $\FF_{q^2}$ of cardinality
$q^2$. Here $\varphi_{d}(x)$ denotes the Chebyshev polynomial of degree $d$.
The same question is studied for the curves corresponding to $y^2=(x\pm 2)
\varphi_{d}(x)$, and also for $y^2=(x^2-4)\varphi_d(x)$.
