This coherent monograph describes and explains quantum phenomena in two-dimensional (2D) electron systems with extremely strong internal interactions, which cannot be described by the conventional Fermi-liquid approach. The central physical objects considered are the 2D Coulomb liquid, of which the average Coulomb interaction energy per electron is much higher than the mean kinetic energy, and the Wigner solid. The text provides a new and comprehensive review of the remarkable properties of Coulomb liquids and solids formed on the free surface of liquid helium and other interfaces. This book is intended for graduate students and researchers in the fields of quantum liquids, electronic properties of 2D systems, and solid-state physics. It includes different levels of sophistication so as to be useful for both theorists and experimentalists. The presentation is largely self-contained, and also describes some instructive examples that will be of general interest to solid-state physicists.