While the emergence and evolution of solar surface magnetic flux reveals what goes on in the solar interior, the interplay of convection and magnetic field in the photosphere regulates the field dispersal and drives the instabilities which heat the outer solar atmosphere. This book presents a synthesis between observers and theorists, both with regard to the magnetic elements which make up solar magnetic fields (ranging from tiny flux tubes to whole active regions), and to the surface patterns in which these elements display properties of the subsurface dynamo. A major breakthrough comes from numerical simulations. Modelling of flux concentration, flux tube dynamics, penumbral toplogy, umbral fine structure, and so on, turns solar physics into an experimental science. The reviews and research papers in this volume provide an overview of the solar frontier of astrophysical magnetohydrodynamics. The elements and patterns of solar surface magnetism contain much information about the subsurface solar dynamo, as well as on the magnetically-dominated energy budget and structuring of the outer solar atmosphere. The volume treats high-resolution solar polarimetry, the physics of solar magnetic elements, and the information contained in their patterns of emergence on the solar surface in depth, with a balance between theoretical and observational studies.