In this communication, the authors survey their version of discrete vector calculus, in which they have been working along the last two decades. The essential difference with other approaches is the introduction of the tangent space at each point of a graph. Once this structure is defined, other ingredients of discrete vector calculus, especially vector fields and endomorphism fields, can be easily and properly introduced. In addition, the definition of the order of a linear operator, relating functions and vector fields, allows us to show the local character of the main difference operators as derivative, gradient, divergence and curl. The analogy with the techniques and approaches of the differentiable case allows us to raise problems that would be difficult to conceive without the previous notions. We show here how boundary problems can be posed, the analysis of which requires analogs of the balance theorems of Mathematical Physics. We define the exterior normal to a (finite) set and obtain the discrete versions of the divergence theorem and Green’s Identities.