  In the first part of the paper we study the structure of Banach spaces with a
conditional spreading basis. The geometry of such spaces exhibit a striking
resemblance to the geometry of James' space. Further, we show that the
averaging projections onto subspaces spanned by constant coefficient blocks
with no gaps between supports are bounded. As a consequence, every Banach space
with a spreading basis contains a complemented subspace with an unconditional
basis. This gives an affirmative answer to a question of H. Rosenthal.
  The second part contains two results on Banach spaces $X$ whose asymptotic
structures are closely related to $c_0$ and do not contain a copy of $\ell_1$:
i) Suppose $X$ has a normalized weakly null basis $(x_i)$ and every spreading
model $(e_i)$ of a normalized weakly null block basis satisfies
$\|e_1-e_2\|=1$. Then some subsequence of $(x_i)$ is equivalent to the unit
vector basis of $c_0$. This generalizes a similar theorem of Odell and
Schlumprecht, and yields a new proof of the Elton-Odell theorem on the
existence of infinite $(1+\varepsilon)$-separated sequences in the unit sphere
of an arbitrary infinite dimensional Banach space. ii) Suppose that all
asymptotic models of $X$ generated by weakly null arrays are equivalent to the
unit vector basis of $c_0$. Then $X^*$ is separable and $X$ is asymptotic-$c_0$
with respect to a shrinking basis $(y_i)$ of $Y\supseteq X$.
