  One of the most promising possibilities may be the appearance of quark matter
in astrophysical phenomena in the light of recent progress in observations. The
mechanism of deconfinement is not well understood, but the thermodynamical
aspects of the hadron-quark (HQ) phase transition have been extensively studied
in recent years. Then the mixed phase of hadron and quark matter becomes
important; the proper treatment is needed to describe the HQ phase transition
and derive the equation of state (EOS) for the HQ matter, based on the Gibbs
conditions for phase equilibrium. We here adopt a EOS based on the
baryon-baryon interactions including hyperons for the hadron phase, while we
use rather simple EOS within the MIT bag model in the quark phase. For quark
matter we further try to improve the previous EOS by considering other
effective models of QCD. One of the interesting consequences may be the
appearance of the inhomogeneous structures called "pasta", which are brought
about by the surface and the Coulomb interaction effects. We present here a
comprehensive review of our recent works about the HQ phase transition in
various astrophysical situations: cold catalyzed matter, hot matter and
neutrino-trapped matter. We show how the pasta structure becomes unstable by
the charge screening of the Coulomb interaction, thermal effect or the neutrino
trapping effect. Such inhomogeneous structure may affect astrophysical
phenomena through its elasticity or thermal properties. Here we also discuss
some implications on supernova explosion, gravitational wave and cooling of
compact stars.
