Involutive bases are a special form of non-reduced Gr bner bases with additional combinatorial properties. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and non-multiplicative Given an admissible monomial ordering, this separation is applied to polynomials in terms of their leading monomials. As special cases of the separation we consider those introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis of partial differential equations Given involutive division, we define an involutive reduction and an involutive normal form. Then we introduce the concept of involutivity for polynomial systems. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain conditions, for example, Janet, provide an algorithmic construction of an involutive basis for any polynomial ideal. Much of the existing literature on involutive bases concentrates on their efficient algorithmic construction. By contrast, we are here more concerned with their structural properties. Pommaret bases are not only important for differential equations, but also define a special type of decomposition, a Rees decomposition. The main topic of the four th chapter is to show that this fact makes them a very powerful tool for computation algebraic geometry. Most of these applications exploit that Pommaret bases possess a highly interesting syzygy theory. For example, they allow for directly reading off the depth, the Krull dimension and for simple constructive proofs of both Hilbert’s Syzygy Theorem. We use these results for simple proofs of Hironaka’s criterion for Cohen-Macaulay modules. We show that the involutive standard representations of the non-multiplicative multiples of the generators induce a Gr bner basis (for an appropriately chosen monomial order) of the first syzygy module. Essentially, this involutive form of Schreyer’s theorem follows from the ideas behind Buchberger’s second criterion for redundant S-polynomials. For Janet and Pommaret bases the situation is even better, as the arising Gr bner basis is then again a Janet and Pommaret basis, respectively. It is well-known that Pommaret bases do not always exist but only in so-called -regular coordinates. We show that several type="#_x0000_t75" -regular coordinates that is more efficient than all methods proposed in the literature so far.