Let
$X,Y$
be Banach spaces. Of concern are the higher order abstract Cauchy problem
$({\rm ACP}_n)$
in
$X$
and its inhomogeneous version
$({\rm IACP}_n)$
. A new operator family of bounded linear operators from
$Y$
to
$X$
is introduced, called an existence family for
$({\rm ACP}_n)$
, so that the existence and continuous dependence on initial data of the solutions of
$({\rm ACP}_n)$
and
$({\rm IACP}_n)$
can be studied, and some basic results in a quite general setting can be obtained. A sufficient and necessary condition ensuring that
$({\rm ACP}_n)$
possesses an exponentially bounded existence family, in terms of Laplace transforms, is presented. As a partner of the existence family, for
$({\rm ACP}_n)$
, a uniqueness family of bounded linear operators on
$X$
is defined to guarantee the uniqueness of solutions. These two operator families for
$({\rm ACP}_n)$
are generalizations of the classical strongly continuous semigroups and sine operator functions, the
$C$
-regularized semigroups and sine operator functions, the existence and uniqueness families for
$(ACP_1)$
, and the
$C$
-propagation families for
$({\rm ACP}_n)$
. They have a special function in treating those ill-posed
$({\rm ACP}_n)$
and
$({\rm IACP}_n)$
whose coefficient operators lack commutativity.