  If $X$ is a manifold then the set $C^\infty(X)$ of smooth functions
$f:X\to\mathbb R$ 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 this book, a sequel to the second author's monograph on
$C^\infty$-algebraic geometry arXiv:1001.0023, 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 as the foundations of theories of derived manifolds and
derived orbifolds with corners.
  This book is based on the PhD thesis of the first author, supervised by the
second author.
