The present paper is a revised version of a much longer
work (which had fewer results) entitled "Block Bundle Sheaves".
This is an obsolete phrase for "stratified polyhedra"; and since
my work has been referred to under the old title, I mention it as
a sort of subtitle of the present work. During the process of
revision, I profited greatly from lecturing in a graduate seminar
at M.I.T., since it was necessary to describe technical definitions
and procedures intuitively. I talked about my difficulties to
many friends, especially Ralph Reid, my sister Ellen and Dennis
Sullivan. Above all, I am indebted to Colin Rourke, who suggested
the present method of defining stratifications, and who urged me
to change terminology to conform better with Thom's theory [2 8]
of "ensembles stratifies". I am also glad to express my gratitude
to William Browder.