We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case, locally and globally in time, with sufficiently small initial data, and for a simple shear flow problem, locally in time with arbitrary initial data; after that we extend our results to some 3D flow problems, locally in time, with large initial data. Additionally, we present results for models of White-Metzner type in 3D flow, locally and globally in time, with sufficiently small initial data. We solve our problem using an iteration between elliptic and hyperbolic linear subproblems. The limit of the iteration provides the solution of our original problem. / Ph. D.