The field of boundary element methods (BEM) relies on recasting boundary value problems for (mostly linear) partial differential equations as (usually singular) integral equations on boundaries of domains or interfaces. Its main goal is the design and analysis of methods and algorithms for the stable and accurate discretization of these integral equations, the data-sparse representation of the resulting systems of equations, and their efficient direct or iterative solution. Boundary element methods play a key role in important areas of computational engineering and physics addressing simulations in acoustics, electromagnetics, and elasticity. Thus progress in boundary element method, both theoretical and algorithmic, is definitely relevant beyond mathematics. Boundary element methods had been developed for many decades, but during the past two decades the field has seen a surge in research activity, spurred by algorithmic and theoretical breakthroughs concerning BEM for electromagnetics, time-domain methods, new approaches to eigenvalue problems, adaptivity, local low-rank matrix compression, and frequency-explicit analysis, to name only a few. The contributions in this report give an impressive panorama of the many and diverse current research activities in BEM. They range profound mathematical analyses with striking results to new algorithmic developments. On the one hand, the results are based on a large variety of tools from many areas of mathematics. On the other hand, research in BEM blazes the trail for progress in the numerical treatment of non-local operators, a field that is rapidly gaining importance.