Is there a definition for vector fields behaving like distributions?

Question

I saw in some physics lecture notes that the Dirac delta function was introduced to model point charges. I read on Wikipedia that it can be defined as a distribution, in other words, a linear functional on test functions. However, I haven't found anyone using vector fields that behave like distributions after a quick search on Google. I expect that there is an object that can be used to model an infinitely large current density passing through an infinitely thin wire, so that we can integrate to find the amount of current passing through a surface. Is there a definition for these objects?