This paper presents a theory of Fourier transforms of complex-valued functions on a finite abelian group and investigates two applications of this theory. Chapter I is an introduction with remarks on notation. Basic theory, including Pontrvagin duality and the Poisson Summation formula, is the subject of Chapter II. In Chapter III the Fourier transform is viewed as an intertwining operator for certain unitary group representations. The solution of the eigenvalue problem of the Fourier transform of functions on the group Z/n of integers module n leads to a proof of the quadratic reciprocity law in Chapter IV. Chapter V addresses the, use of the Fourier transform in computing.