We want to study the derivation and some basic features of the equations of motion, including the Euler equation and the Navier-Stokes equations, that govern the dynamics of fluids. We will discuss the linear and nonlinear stability of various basic fluid flows including convection and shear flows. Lastly, we are interested in turbulent flows where a significant amount of the energy is transferred to small-scale eddies. If time permits, we will also discuss uniqueness and regularity properties of the Navier-Stokes equations and rigorous upper-bound theory that relates certain physical parameters of fluids to certain measures of turbulence.