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Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) wasproposed as a classically equivalent way to impose the infinite number ofWheeler -- DeWitt constraint equations in terms of a single Master Equation.While the proposal has some promising abstract features, it was until nowbarely tested in known models. In this series of five papers we fill this gap,thereby adding confidence to the proposal. We consider a wide range of modelswith increasingly more complicated constraint algebras, beginning with a finitedimensional, Abelean algebra of constraint operators which are linear in themomenta and ending with an infinite dimensional, non-Abelean algebra ofconstraint operators which closes with structure functions only and which arenot even polynomial in the momenta. In all these models we apply the MasterConstraint Programme successfully, however, the full flexibility of the methodmust be exploited in order to complete our task. This shows that the MasterConstraint Programme has a wide range of applicability but that there are many,physically interesting subtleties that must be taken care of in doing so. Inthis first paper we prepare the analysis of our test models by outlining thegeneral framework of the Master Constraint Programme. The models themselveswill be studied in the remaining four papers. As a side result we develop theDirect Integral Decomposition (DID) for solving quantum constraints as analternative to Refined Algebraic Quantization (RAQ).Comment: 42 pages, no figure

Testing the master constraint programme for loop quantum gravity: I. General framework