A notion of regular singularity for stratified bundles on a smooth variety over an algebraically closed field of positive characteristic is introduced and studied. Special emphasis is put on the relation of such objects with the étale fundamental group of the variety. A major result of this thesis is that a stratified bundle is regular singular with finite monodromy if and only if it is trivialized on a finite tame étale covering. This theorem provides a precise link between regular singular differential equations and tame ramification. It is also studied in which sense the (tame) étale fundamental group controls the category of regular singular stratified bundles. Conjecturally, if the tame étale fundamental group is trivial, then so is the category of regular singular stratified bundles. Partial results in this direction are proved in the final chapter of this thesis. Four appendices collecting background information are included for the reader's convenience.