Despite the linearity of the Stokes equations, nonlinearity is prevalent at zero Reynolds number due to the presence of interfaces, elastic structures and polymers, chemical concentrations, and electric fields, all of which have ramifications across many industrial and biological processes. The dual focus of this Colloquium will be on nonlinearities involving slender, elastic filaments and chemically-active particles. Attention will be given to the dynamics of individual filaments and particles, as well as their collective motion. We aim for the Colloquium to bring together theoreticians and experimentalists in these exciting areas to explore their commonalities and differences, and facilitate an exchange of techniques and methodologies.