  We define a Heegaard-Scharlemann-Thompson (HST) splitting of a 3-manifold M
to be a sequence of pairwise-disjoint, embedded surfaces, {F_i}, such that for
each odd value of i, F_i is a Heegaard splitting of the submanifold of M
cobounded by F_{i-1} and F_{i+1}. Our main result is the following: Suppose M
(\neq B^3 or S^3) is an irreducible submanifold of a triangulated 3-manifold,
bounded by a normal or almost normal surface, and containing at most one
maximal normal 2-sphere. If {F_i} is a strongly irreducible HST splitting of M
then we may isotope it so that for each even value of i the surface F_i is
normal and for each odd value of i the surface F_i is almost normal.
  We then show how various theorems of Rubinstein, Thompson, Stocking and
Schleimer follow from this result. We also show how our results imply the
following: (1) a manifold that contains a non-separating surface contains an
almost normal one, and (2) if a manifold contains a normal Heegaard surface
then it contains two almost normal ones that are topologically parallel to it.
