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.
Categories: math.DG math-ph math.MP