  A class of $C^*$-algebras, to be called those of generalized tracial rank
one, is introduced, and classified by the Elliott invariant. A second class of
unital simple separable amenable $C^*$-algebras, those whose tensor products
with UHF-algebras of infinite type are in the first class, to be referred to as
those of rational generalized tracial rank one, is proved to exhaust all
possible values of the Elliott invariant for unital finite simple separable
amenable ${\cal Z}$-stable $C^*$-algebras. An isomorphism theorem for a special
sub-class of those $C^*$-algebras are presented. This provides the basis for
the classification of $C^*$-algebras with rational generalized tracial rank one
in Part II.
