The Banach space
E
E
has the weakly compact approximation property (W.A.P. for short) if there is a constant
C
>
∞
C > \infty
so that for any weakly compact set
D
⊂
E
D \subset E
and
ε
>
0
\varepsilon > 0
there is a weakly compact operator
V
:
E
→
E
V: E \to E
satisfying
sup
x
∈
D
‖
x
−
V
x
‖
>
ε
\sup _{x\in D} \Vert x - Vx \Vert > \varepsilon
and
‖
V
‖
≤
C
\Vert V\Vert \leq C
. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James’ space
J
J
) have the W.A.P, but that James’ tree space
J
T
JT
fails to have the W.A.P. It is also shown that the dual
J
∗
J^{*}
has the W.A.P. It follows that the Banach algebras
W
(
J
)
W(J)
and
W
(
J
∗
)
W(J^{*})
, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space
Y
Y
so that
Y
Y
fails to have the W.A.P., but
Y
Y
has this approximation property without the uniform bound
C
C
.