We develop a least-squares finite element method for linear
Partial Differential Equations (PDEs) of hyperbolic type. This formulation
allows for discontinuities in the numerical approximation and yields a
linear system which can be handled efficiently by Algebraic
We pose the classical advection equation as a ``dual''-type problem and relate the formulation to previous attempts. Convergence properties and solution quality for discontinuous solutions are investigated for standard, conforming finite element spaces on quasi-uniform tessellations of the domain and for the general case when the flow field is not aligned with the computational mesh. Algebraic Multigrid results are presented for the linear system arising from the discretization and we study the success of this solver.