  We construct several new classes of transcendental entire functions, f, such
that both the escaping set, I(f), and the fast escaping set, A(f), have a
structure known as a spider's web. We show that some of these classes have a
degree of stability under changes in the function. We show that new examples of
functions for which I(f) and A(f) are spiders' webs can be constructed by
composition, by differentiation, and by integration of existing examples. We
use a property of spiders' webs to give new results concerning functions with
no unbounded Fatou components.
