This paper is devoted to the presentation of a general patchworking procedure to construct reduced singular complex curves having prescribed singularities and belonging to a given linear system on an algebraic surface. The patchworking procedure to construct a curve \(C\) with singularities of given type on an algebraic surface \(X\) uses a reducible surface \(X_0\) which is a degeneration of \(X\), then construct a curve \(C_0\) on \(X_0\) and prove that \(C_0\) deforms to \(C\) on \(X\). The paper under review significantly generalizes the preceding works by the authors and others by using weaker assumptions to prove that \(C_0\) deforms to \(C\) on \(X\). In the second part of this paper [Isr. J. Math. 151, 145--166 (2006; Zbl 1128.14020)], the authors apply this general procedure to produce detailed examples.