  We review techniques for simulating fully quantum nonadiabatic dynamics using
the frozen-width moving Gaussian basis functions to represent the nuclear
wavefunction. A choice of these basis functions is primarily motivated by the
idea of the on-the-fly dynamics that will involve electronic structure
calculations done locally in the vicinity of each Gaussian center and thus
avoiding the "curse of dimensionality" appearing in large systems. For quantum
dynamics involving multiple electronic states there are several aspects that
need to be addressed. First, the choice of the electronic state representation
is one of most defining in terms of formulation of resulting equations of
motion. We will discuss pros and cons of the standard adiabatic and diabatic
representations as well as the relatively new moving crude adiabatic
representation. Second, if the number of electronic states can be fixed
throughout the dynamics, the situation is different for the number of Gaussians
needed for an accurate expansion of the total wavefunction. The latter
increases its complexity along the course of the dynamics and a protocol
extending the number of Gaussians is needed. We will consider two common
approaches for the extension: 1) spawning and 2) cloning. Third, equations of
motion for individual Gaussians can be chosen in different ways, implications
for the energy conservation related to these ways will be discussed. Finally,
to extend the success of moving basis approaches to quantum dynamics of open
systems we will consider the Non-stochastic Open System Schr\"odinger Equation
(NOSSE).
