Math. Proc. Cambridge Philos. Soc. 151, 3 (2011), 551–571. Also available on the arXiv. We construct several new classes of transcendental entire functions, , such that both the escaping set, , and the fast escaping set, , 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 and 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.