  In this paper, we give necessary and sufficient conditions for the existence
of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we
show that for every integer $r\ge 2$ there exists a family of indecomposable
rank $r$ Ulrich bundles on the certain cubic fourfolds, depending roughly on
$r$ parameters, and in particular they are of wild representation type; special
surfaces on the cubic fourfolds are explicitly constructed by Macaulay2; a new
$19$-dimensional family of projective ten-dimensional irreducible holomorphic
symplectic manifolds associated to a certain cubic fourfold is constructed; and
for certain cubic fourfold $X$, there exist arithmetically Cohen-Macaulay
smooth surface $Y \subset X$ which are not an intersection $X \cap T$ for a
codimension two subvariety $T \subset \Bbb P^5$.
