Variational inequalities are applied to formulate and to examine Compactness Property of anisotropic Hele-Shaw flows. A new concept Weighted Elliptic Quadrature Domains has been introduced to facilitate the proof. The author also generalized some techniques in studying Hele-Shaw flows to abstract settings. We obtained $H\sp{2,2}$ regularity of solutions for variational inequalities associated with p-harmonic operators. For a class of variational inequalities associated with non-coercive monotone operators, we established a necessary and sufficient condition for the existence of a solution and applied it to a penalization procedure. By using Caffarelli's result, we showed that the Hele-Shaw moving boundary (isotropic case) can be locally represented as $\{$x; x = $\nabla H$(x)$\}$ for some harmonic function H.