We study new categories of ’ $C^\infty$-schemes with corners’, following the definition of schemes in Algebraic Geometry. A ‘(local) $C^\infty$-ringed space with corners’ is a topological space with a sheaf of $C^\infty$-rings with corners. We define a spectrum functor from $C^\infty$-rings with corners to local $C^\infty$-ringed spaces with corners, adjoint to the global sections functor. Then a $C^\infty$-scheme with corners is a local $C^\infty$-ringed space with corners which is locally isomorphic to the spectrum of a $C^\infty$-ring with corners.

Manifolds with (g-)corners embed into $C^\infty$-schemes with corners as full subcategories. Thus, $C^\infty$-schemes with corners are a vast generalization of manifolds with corners and manifolds with g-corners.

In ordinary Algebraic Geometry, the spectrum functor is full and faithful, but for $C^\infty$-rings with corners neither holds. ‘Semi-complete’ $C^\infty$-rings with corners are a special class on which the spectrum functor is faithful and surjective, but not full.

We study special classes of $C^\infty$-schemes with corners, and existence of fibre products of $C^\infty$-schemes with corners.