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Our starting point is the following pathology present in all ‘classical’ geometries such as algebraic geometry, formal algebraic geometry, analytic geometry, differential geometry, etc. Assume X is an object of such a geometry and assume we are given a correspondence on X, i.e., another object X^\prime of this geometry together with two morphisms from X^\prime to X. Such a correspondence generates an equivalence relation on X and one would like to construct, in our geometry, an object that plays the role of the quotient of X by this equivalence relation. However, in many interesting cases, the orbits of the equivalence relation are dense in X in which case the categorical quotient is trivial. One way out of this puzzle would be to consider a refinement of the original geometry by allowing more functions on our objects in order to achieve the ‘separation of the orbits’. An example of such a move is Alain Connes’ invention of noncommutative geometry [Con] which led to a refinement of differential geometry and to the construction of new quotients in a series of remarkable examples. In a different direction, a refinement of algebraic geometry was proposed in a series of papers beginning with [Bu95a] (and partially summarized in [Bu05]); this refined geometry can be referred to as δ-geometry and is roughly obtained from the usual algebraic geometry by replacing unital commutative rings with δ-rings, i.e., with {\mathbb {Z}}_{(p)}{\mathbb {Z}}_{(p)}{\mathbb {Z}}_{(p)}{\mathbb {Z}}_{(p)})-algebras that admit (and are equipped with) a Fermat quotient operator (which we denote by δ and we refer to as a p-derivation). The latter led in [Bu05] to the construction of quotients of various curves X (rational, elliptic, modular) by various remarkable correspondences. The aim of the present monograph is twofold. First we revisit the foundations of δ-geometry by offering a version of it which is more flexible and more general than that of [Bu05]. Second, as our main application, we construct quotients for a remarkable series of higher dimensional examples that generalize the modular curve examples in [Bu05]. The method offered in this monograph is general in nature but we concentrate on a foundational case study which is that of quotients of moduli schemes of principally polarized abelian schemes of arbitrary fixed relative dimension by Hecke correspondences and which we hope it will be used for all future generalizations to closed subschemes of these moduli schemes that are defined by integral canonical models of Shimura varieties of Hodge type (cf. [V], Chap. 1, Sect. 1.6). Our treatment here is independent of that in [Bu05] and, in fact, all our prerequisites here belong to ‘usual’ algebraic geometry: we mainly assume some familiarity with abelian schemes and their deformation theory for which we provide detailed references. A key technical role in our proofs here is played by some old and new results in the classical geometric invariant theory of tuples of quadrics. These are carefully reviewed and developed in the last chapter of our monograph. With some rare exceptions, all results in this monograph appear here for the first time. It is interesting to note that some of the examples of quotient problems for which δ-geometry is effective are related to examples for which noncommutative geometry proves to be effective. We also note that a variant of δ-geometry was used in [Borg] as a candidate for the geometry over {\mathbb {F}}_{1}{\mathbb {F}}_{1}, the field with one element (see also [Bu05], Introd.); on the other hand, noncommutative geometry has also been proposed as a path towards {\mathbb {F}}_{1}{\mathbb {F}}_{1}. Finally note that δ-rings recently played a role in the foundation of prismatic cohomology (see [BS]). It would be interesting to interrelate these topics and the topic of the present monograph.