  We investigate the properties of mixed stars formed by hadronic and quark
matter in $\beta$-equilibrium described by appropriate equations of state (EOS)
in the framework of relativistic mean-field theory. We use the non- linear
Walecka model for the hadron matter and the MIT Bag and the Nambu-Jona-Lasinio
models for the quark matter. The phase transition to a deconfined quark phase
is investigated. In particular, we study the dependence of the onset of a mixed
phase and a pure quark phase on the hyperon couplings, quark model and
properties of the hadronic model. We calculate the strangeness fraction with
baryonic density for the different EOS. With the NJL model the strangeness
content in the mixed phase decreases. The calculations were performed for T=0
and for finite temperatures in order to describe neutron and proto-neutron
stars. The star properties are discussed. Both the Bag model and the NJL model
predict a mixed phase in the interior of the star. Maximum allowed masses for
proto-neutron stars are larger for the NJL model ($\sim 1.9$ M$_{\bigodot}$)
than for the Bag model ($\sim 1.6$ M$_{\bigodot}$).
