The well-posedness of nonsmooth semi-explicit differential-algebraic equations (DAEs) is investigated (click here; free access is provided until February 08, 2017). The nonsmooth DAEs are classified as having differential index one in a generalized sense. Existence of solutions is derived under consistency and regularity of the initial data. Uniqueness of a solution is guaranteed under analogous nonsmooth ordinary-differential equation uniqueness assumptions. The continuation of solutions is established and sufficient conditions for continuous and Lipschitzian parametric dependence of solutions are also provided. The findings here are a natural extension of classical results and lay the foundation for further theoretical and computational analyses of nonsmooth DAEs.