  The traditional approach of defining the fundamental group first and then
constructing universal coverings works well only for the class of Poincar\' e
spaces. For general spaces there were several attempts to define generalized
coverings (see \cite{BP3}, \cite{BDLM}, and \cite{FisZas}), yet there is no
general theory so far that covers all path connected spaces. In this paper we
plan to remedy that by changing the order of things: we define the universal
covering first and its group of deck transformations is the new fundamental
group of the base space.
  The basic idea is that a non-trivial loop ought to be detected by a covering
(not by extension over the unit disk): a loop is non-trivial if there is a
covering such that some lift of the loop is a non-loop.
  So it remains to define coverings: the most natural class is the class of
maps that have unique disk lifting property. To make the theory work one needs
to add the assumption that path components of pre-images of open sets form a
basis of the total space.
