  We introduce the notion of a C*-valued weight between two C*-algebras as a
generalization of an ordinary weight on a C*-algebra and as a C*-version of
operator valued weights on von Neumann algebras. Also, some form of lower
semi-continuity will be discussed together with an extension to the multiplier
algebra. A strong but useful condition for C*-valued weights, the so-called
regularity, is introduced. At the same time, we propose a construction
procedure for such regular C*-valued weights. This construction procedure will
be used to define the tensor product of regular C*-valued weights.
