  Let X be a smooth variety over an algebraically closed field k of positive
characteristic. We define and study a general notion of regular singularities
for stratified bundles (i.e. O_X-coherent D_X-modules) on X without relying on
resolution of singularities. The main result is that the category of regular
singular stratified bundles with finite monodromy is equivalent to the category
of continuous representations of the tame fundamental group on finite
dimensional k-vector spaces. As a corollary we obtain that a stratified bundle
with finite monodromy is regular singular if and only if it is regular singular
along all curves mapping to X.
