  This paper is the fifth and final in a series on embedded minimal surfaces.
Following our earlier papers on disks, we prove here two main structure
theorems for non-simply connected embedded minimal surfaces of any given fixed
genus.
  The first of these asserts that any such surface without small necks can be
obtained by gluing together two oppositely--oriented double spiral staircases.
  The second gives a pair of pants decomposition of any such surface when there
are small necks, cutting the surface along a collection of short curves. After
the cutting, we are left with graphical pieces that are defined over a disk
with either one or two sub--disks removed (a topological disk with two
sub--disks removed is called a pair of pants).
  Both of these structures occur as different extremes in the two-parameter
family of minimal surfaces known as the Riemann examples.
  The results of [CM3]--[CM6] have already been used by many authors; see,
e.g., the surveys [MeP], [P], [Ro] and the introduction in [CM6] for some of
these applications. There is much current research on minimal surfaces with
infinite topology. Some of the results of the present paper were announced
previously and have already been widely used to study infinite topology minimal
surfaces; see, e.g., [MeP], [MePRs1], [MePRs2], [MePRs3], and [P].
