Math. Proc. Cambridge Philos. Soc. 151, 3 (2011), 551–571. Also available on the arXiv.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 web’s can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders web’s to give new results concerning functions with no unbounded Fatou components.