  We introduce the higher order spreading models associated to a Banach space
$X$. Their definition is based on $\ff$-sequences $(x_s)_{s\in\ff}$ with $\ff$
a regular thin family and the plegma families. We show that the higher order
spreading models of a Banach space $X$ form an increasing transfinite hierarchy
$(\mathcal{SM}_\xi(X))_{\xi<\omega_1}$. Each $\mathcal{SM}_\xi (X)$ contains
all spreading models generated by $\ff$-sequences $(x_s)_{s\in\ff}$ with order
of $\ff$ equal to $\xi$. We also provide a study of the fundamental properties
of the hierarchy.
