In this paper, we study the duality theory of Hopf C -algebras in a general "Hilbert-space-free" framework. Our particular interests are the "full duality" and the "reduced duality". In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf C*-algebra. This study of reduced duality and Fourier algebra is found to be useful in the study of other aspects of Hopf C -algebras (see e.g. [12], [13] and [14]).