  A theory of graded manifolds can be viewed as a generalization of
differential geometry of smooth manifolds. It allows one to work with functions
which locally depend not only on ordinary real variables, but also on
$\mathbb{Z}$-graded variables which can either commute or anticommute,
according to their degree. To obtain a consistent global description of graded
manifolds, one resorts to sheaves of graded commutative associative algebras on
second countable Hausdorff topological spaces, locally isomorphic to a suitable
"model space".
  This paper aims to build robust mathematical foundations of geometry of
graded manifolds. Some known issues in their definition are resolved,
especially the case where positively and negatively graded coordinates appear
together. The focus is on a detailed exposition of standard geometrical
constructions rather then on applications. Necessary excerpts from graded
algebra and graded sheaf theory are included.
