  If $\psi$ is analytic on the open unit disk $\mathbb{D}$ and $\varphi$ is an
analytic self-map of $\mathbb{D}$, the weighted composition operator
$C_{\psi,\varphi}$ is defined by $C_{\psi,\varphi}f(z)=\psi(z)f (\varphi (z))$,
when $f$ is analytic on $\mathbb{D}$. In this paper, we study normal,
cohyponormal, hyponormal and normaloid weighted composition operators on the
Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces
$H^{2}(\beta)$, we prove that if $C_{\psi,\varphi}$ is cohyponormal on
$H^{2}(\beta)$, then $\psi$ never vanishes on $\mathbb{D}$ and $\varphi$ is
univalent, when $\psi \not \equiv 0$ and $\varphi$ is not a constant function.
Moreover, for $\psi=K_{a}$, where $|a| < 1$, we investigate normal,
cohyponormal and hyponormal weighted composition operators $C_{\psi,\varphi}$.
After that, for $\varphi $ which is a hyperbolic or parabolic automorphism, we
characterize all normal weighted composition operators $C_{\psi,\varphi}$, when
$\psi \not \equiv 0$ and $\psi$ is analytic on $\overline{\mathbb{D}}$.
  Finally, we find all normal weighted composition operators which are bounded
below.
