Accurately and efficiently simulating the quantum dynamics of an open quantum system remains a challenging task in modern quantum physics and chemical physics, due to the unfavorable exponential scaling of the quantum problem. To resolve this challenge, mixed quantum-classical (MQC) approaches have been developed by treating the crucial part of the system as the quantum subsystem, and the other parts as classical degrees of freedom (DOFs). This type of approach is particularly successful for describing non-adiabatic molecular dynamics that involve electronic-nuclear interactions, given the fact that the nuclear DOFs are in general anharmonic and their interactions with the electronic DOFs are non-Markovian, thus rendering many approximate master equations not directly applicable. One major category of these mixed quantum-classical methods is surface hopping, most notably Tully's fewest-switches surface hoping approach and several later additions to this method. Another major category is the mean-field Ehrenfest (MFE) approach, or more generally, semiclassical mapping approaches based on the Meyer-Miller mapping formalism. All of these approaches explicitly propagate the classical DOFs and captures their influence on the quantum DOFs through the parametric dependence of the quantum equations of motion (EOM) on the classical trajectories.