Rational homotopy theory for non-simply connected spaces

We construct an algebraic rational homotopy theory for all connected CW spa ces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simpl y connected case and has two basic properties: (1) it induces a natural equ ivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebrais ation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.