We study the thermodynamics and phase diagrams of two-flavor quantum chromodynamics using the Polyakov-loop extended quark-meson (PQM) model and the Pisarski-Skokov chiral matrix ($\chi M$) model. At temperatures up to $T\approx2T_c$ and baryon chemical potentials up to $\mu_B=400\ \rm{MeV}$, both models show reasonable agreement with the pressure, energy density, and interaction measure as calculated on the lattice... The Polyakov loop is found to rise significantly faster with temperature in models than on the lattice. In the low-temperature and high baryon density regime, the two models predict different states of matter; The PQM model predicts a confined and chirally restored phase, while the $\chi M$ model predicts a deconfined and chirally restored phase. At finite isospin density and zero baryon density, the onset of pion condensation at $T=0$ is at $\mu_I={1\over2}m_{\pi}$, and the transition is second order at all temperatures. The transition temperature for pion condensation coincides with that of the chiral transition for values of the isospin chemical potential larger than approximately $110\ \rm{MeV}$. In the $\chi M$ model they also coincide with the transition temperature for deconfinement. The results are in good overall agreement with recent lattice simulations of the $\mu_I$--$T$ phase diagram. (read more)