  We present a comprehensive review of the nodal domains and lines of quantum
billiards, emphasizing a quantitative comparison of theoretical findings to
experiments. The nodal statistics are shown to distinguish not only between
regular and chaotic classical dynamics but also between different geometric
shapes of the billiard system itself. We discuss, in particular, how a random
superposition of plane waves can model chaotic eigenfunctions and highlight the
connections of the complex morphology of the nodal lines thereof to percolation
theory and Schramm-Loewner evolution. Various approaches to counting the nodal
domains---using trace formulae, graph theory, and difference equations---are
also illustrated with examples. The nodal patterns addressed pertain to waves
on vibrating plates and membranes, acoustic and electromagnetic modes,
wavefunctions of a "particle in a box'" as well as to percolating clusters, and
domains in ferromagnets, thus underlining the diversity---and far-reaching
implications---of the problem.
