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This article is devoted to completing some aspects of the classicalCauchy-Lipschitz (or Picard-Lindel\"of) theory for general nonlinear systemsposed on time scales, that are closed subsets of the set of real numbers.Partial results do exist but do not cover the framework of general dynamics ontime-scales encountered e.g. in applications to control theory. In the presentwork, we first introduce the notion of absolutely continuous solution forshifted and non shifted Delta-Cauchy problems, and then the notion of a maximalsolution. We state and prove a Cauchy-Lipschitz theorem, providing existenceand uniqueness of the maximal solution of a given Delta-Cauchy problem undersuitable assumptions like regressivity and local Lipschitz continuity, anddiscuss some related issues like the behavior of maximal solutions at terminalpoints.

General Cauchy-Lipschitz theory for shifted and non shifted Delta-Cauchy problems on time scales