Turbulent fluid flow is governed by the Navier-Stokes equations. Because of the difficulty of working with the Navier-Stokes equations, several different approximations of the Navier-Stokes equations have been developed. One recently derived approximation is the Lagrangian Averaged Navier-Stokes equations. This thesis will focus on three main areas. First, we seek local solutions to the Lagrangian Averaged Navier-Stokes equations with initial data in Sobolev space with minimal regularity. In some special cases, these local solutions can be extended to global solutions. Secondly, we seek solutions to the Lagrangian Averaged Navier-Stokes equations for initial data in Besov spaces. Finally, we get a global result for Besov spaces in the p = 2 case and a qualitatively different local result for general p by using a different technique.