  Stochastic webs were discovered, first by Arnold for multi-dimensional
Hamiltonian systems, and later by Chernikov et al. for the low-dimensional
case. Generated by weak perturbations, they consist of thread-like regions of
chaotic dynamics in phase space. Their importance is that, in principle, they
enable transport from small energies to high energies. In this introductory
review, we concentrate on low-dimensional stochastic webs and on their
applications to quantum transport in semiconductor superlattices subject to
electric and magnetic fields. We also describe a recently-suggested
modification of the stochastic web to enhance chaotic transport through it and
we discuss its possible applications to superlattices. keywords: stochastic
webs; quantum transport; superlattices; separatrix chaos
