The manuscript serves as a prelude to and presents new and recent advances and a formal theory describing a unified framework towards the generic design of time discretized operators for structural dynamics applications. Emphasis is placed on both linear and subsequently the associated nonlinear applications and implementation aspects. Particular attention is paid to: 1)generic design framework and development of computational algorithms including new avenues which have not been explored to-date, 2)pr oviding a common platform based on stability (unconditional stability required)and accuracy (at least second-order accuracy required), and overshooting behavior and therein assessing algorithmic attributes as related to numerical dissipation and dispersion, and 3)implementation aspects and computational simplicity. The resulting and so-called generalized integration operators encompass a wide vriety of new computational algorithms which have not been explored to-date including the recovery of various existing algorithms, and primitive building blocks are identified which constitute the overall unified framework. Finally, emanating from a generalized time weighted philosophy, the so-called algorithmic markers serve to provide a basis for classification, characterization and design of computational algorithms.