Hilbert-Schmidt frame(HS-frame) is essentially an operator-valued frame, it is more general than g-frames, and thus, covers some generalizations of frames. This paper addresses the Hilbert-Schmidt frames theory for Hilbert spaces. We first introduce the notion of HS-preframe operator, and characterize the HS-frames, Parseval HS-frames, HS-Riesz bases, HS-orthonormal bases and dual HS-frames in terms of HS-preframe operators. In particular, we characterize the dual HS-frames in a constructive way, that is, the algebraic formulae for all dual HS-frames of a given HS-frame are given. Then we discuss the sum of HS-frames through the properties of HS-preframe operators. Finally, we present the representations of HS-frames in terms of linear combinations of simpler ones such as HS-orthonormal bases, HS-Riesz bases and Parseval HS-frames, especially an HS-frame can be represented as a linear combination of two HS-orthonormal bases if and only if it is an HS-Riesz basis.