In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibits 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 X whose asymptotic structures are closely related to c 0 c_0 and do not contain a copy of ℓ 1 \ell _1 : i) Suppose X X has a normalized weakly null basis ( x i ) (x_i) and every spreading model ( e i ) (e_i) of a normalized weakly null block basis satisfies ‖ e 1 − e 2 ‖ = 1 \|e_1-e_2\|=1 . Then some subsequence of ( x i ) (x_i) is equivalent to the unit vector basis of c 0 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 + ε ) (1+\varepsilon ) -separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space. ii) Suppose that all asymptotic models of X X generated by weakly null arrays are equivalent to the unit vector basis of c 0 c_0 . Then X ∗ X^* is separable and X X is asymptotic- c 0 c_0 with respect to a shrinking basis ( y i ) (y_i) of Y ⊇ X Y\supseteq X .