  This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds
theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give
some of their properties. The requirement to work with a completed graded
symmetric algebra to define functions is made clear. Moreover, we define vector
fields and exhibit their graded local basis. The paper also reviews some
correspondences between differential Z-graded manifolds and algebraic
structures.
