The index of an implicit system of differential equations, also known as a differential algebraic equation or DAE, measures how far, in some sense, the system is from being an explicit ordinary differential equation. There is interest in many areas of applications in working directly with the implicit models that often arise. This fundamental paper summarizes what is known about changing the index and introduces an important technique for index reduction that shows great promise in the numerical solution of certain types of implicit differential equations. In particular, this paper shows an equivalence between semi-explicit and fully implicit DAEs so that the theory for the former applies to the latter of one lower index. Techniques are presented for the reduction of the index without introducing additional constants of integration. The relationship between general DAEs and semi-explicit DAEs is also discussed.