We consider a large class of quasilinear second order elliptic systems of the form       - ∑α,β=1N aαβ(x,u(x)),∇u(x))∂2αβu(x) + b(x,u(x),∇u(x)) = 0, where x varies in an unbounded domain Ω of the Euclidean space RN and u = (u1,...,um) is a vector of functions. These systems generate operators acting between the Sobolev spaces W2,p(Ω, Rm) and Lp(Ω, Rm) for p > N. We investigate then the Fredholm and properness properties of these operators and the connections between them. These functional properties play important roles in the existence theory of nonlinear differential equations, and they are related to two recent topological degrees. A first part of this work is an extension of recent results obtained by Rabier and Stuart who studied the scalar case (a single equation) on RN. Our results cover the case of several equations (coupled equations) defined on more general domains. We also study the question of exponential decay of solutions. The general results obtained in our framework are then applied to more specific and new situations: steady reaction-diffusion systems and nonlinear elasticity, where by means of the topological degree, we prove new existence and global continuation results.