We discuss four generalizations and applications of $C^\infty$-schemes with corners.

1. 1. To provide a model for a theory of ‘Synthetic Differential Geometry with corners’.

2. 2. To a theory of ‘ $C^\infty$-stacks with corners’. $C^\infty$-Stacks are studied in D. Joyce, ‘Algebraic Geometry over $C^\infty$-rings’, Memoirs of the AMS, 2019. Most of the theory extends to corners with no changes.

3. 3. To a theory of ‘ $C^\infty$-schemes with a-corners’. ‘Manifolds with a-corners’ are introduced in D. Joyce, arXiv:1605.05913 as a class of manifolds with corners with an alternative smooth structure, that has applications in analysis, e.g. Morse theory moduli spaces should really be manifolds with a-corners, not corners.

4. 4. To a theory of ‘derived $C^\infty$-schemes and derived $C^\infty$-stacks with corners’, and within these, ‘derived manifolds with corners’ and ‘derived orbifolds with corners’, where ‘derived’ is in the sense of Derived Algebraic Geometry.

If X is a manifold then the set $C^\infty$(X) of smooth functions on X is a ’ $C^\infty$-ring’, a rich algebraic structure with many operations. ‘ $C^\infty$-schemes’ are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry. They include smooth manifolds, but also many singular and infinite-dimensional spaces. They have applications to Synthetic Differential Geometry, and to ‘derived manifolds’ in Derived Differential Geometry.

Manifolds with corners, such as a triangle or a cube, are generalizations of manifolds, with boundaries and corners. They occur in many places in Differential Geometry. In this book we define and study new categories of ‘ $C^\infty$-rings with corners’ and ‘ $C^\infty$-schemes with corners’, which generalize manifolds with corners in the same way that $C^\infty$-rings and $C^\infty$-schemes generalize manifolds. These will be used in future work by the second author as the foundations of theories of derived manifolds and derived orbifolds with corners. These have important applications in Symplectic Geometry, as moduli spaces of pseudo-holomorphic curves should be derived orbifolds with corners.