  In this paper we introduce the local Nori fundamental group scheme of a
reduced scheme or algebraic stack over a perfect field $k$. We give particular
attention to the case of fields: to any field extension $K/k$ we attach a
pro-local group scheme over $k$. We show how this group has many analogies, but
also some crucial differences, with the absolute Galois group. We propose two
conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar
Conjecture, providing some evidence in their favor. Finally we show that the
local fundamental group of a normal variety is a quotient of the local
fundamental group of an open, of its generic point (as it happens for the
\'etale fundamental group) and even of any smooth neighborhood.
