  We define and study asymptotically symmetric Banach spaces (a.s.) and its
variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is
a.s. then all spreading models of X are uniformly symmetric. We show that the
converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent
properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if
X is separable and has the property that every normalized weakly null sequence
in X has a subsequence equivalent to the unit vector basis of c_0 then X is
w.a.s.. We obtain an analogous result if c_0 is replaced by ell_1 and also show
it is false if c_0 is replaced by ell_p, 1 < p < infinity. We prove that if 1
less than or equal p < infinity and the norm of the sum of (x_i)_1^n is of the
order n^{1/p} for all (x_i)_1^n in the n^{th} asymptotic structure of $X$, then
X contains an asymptotic ell_p, hence w.a.s. subspace.
