We review techniques for simulating fully quantum nonadiabatic dynamics using the frozen-width moving Gaussian basis functions to represent the nuclear wave function. 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 (MCA) 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 wave function. 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 Nonstochastic Open System Schrödinger Equation (NOSSE).