The authors present multimatrix generalizations of some well-known orthogonal rank factorizations and show how the idea of a QR- decomposition (QRD), a URV- decomposition (URVD), and a singular value decomposition (SVD), which has become an important tool in the analysis and numerical solution of numerous problems, especially since the development of numerically robust algorithms by Golub and his coworkers, for one matrix can be generalized to any number of matrices of compatible dimensions. Their main idea is based on the reducing of the set of matrices \(A_{1}, A_{2},\ldots,A_{n}\) to a simpler form using unitary transformations only. Hereby, they avoid explicit products and inverses of the matrices that are involved and show that these generalized QR-decompositions (GQRD) can be considered as a preliminary reduction for any generalized singular value decomposition (GSVD). The authors discuss in detail the structure of these generalizations and their relations and give a constructive proof for the generalized QR- decompositions. While all results of the paper are stated for complex matrices, they can be specialized to the real case without many difficulties. It can be done in much the same way as with the SVD for complex and real matrices. In particular, it suffices to restate most results using the term real orthogonal instead of unitary and to replace a superscript ''*'' (which denotes the complex conjugate transpose of a matrix) by a superscript ''T'' (which is the transpose of a matrix).