  We extend the concept of weaving Hilbert space frames to the Banach space
setting. Similar to frames in a Hilbert space, we show that for any two
approximate Schauder frames for a Banach space, every weaving is an approximate
Schauder frame if and only if there is a uniform constant $C\geq 1$ such that
every weaving is a $C$-approximate Schauder frame. We also study weaving
Schauder bases, where it is necessary to introduce two notions of weaving. On
one hand, we can ask if two Schauder bases are woven when considered as
Schauder frames with their biorthogonal functionals, and alternatively, we can
ask if each weaving of two Schauder bases remains a Schauder basis. We will
prove that these two notions coincide when all weavings are unconditional, but
otherwise they can be different. Lastly, we prove two perturbation theorems for
approximate Schauder frames.
