  The main purposes of this article are to extend our previous results on
homogeneous sprays to arbitrary (generalized) sprays, to show that locally
diffeomorphic exponential maps can be defined for any (generalized) spray, and
to give a (possibly nonlinear) covariant derivative for any (possibly
nonlinear) connection. In the process, we introduce vertically homogeneous
connections. Unlike homogeneous connections, these allow us to include Finsler
spaces among the applications.
  We provide significant support for the prospect of studying nonlinear
connections via (generalized) sprays. One of the most important is our
generalized APS correspondence.
