The occasion of Enrique Artal's and Alejandro Melle's birthday celebration, held in Jaca, Spain, on June 16-23, 2023, provided the backdrop for a gathering of world-renowned specialists, young researchers, and graduate students interested in singularity theory from the point of view of its multiple connections with the fields of topology, algebra, geometry, and number theory. This volume collects research papers and surveys from speakers and participants, together presenting a panorama of cutting-edge problems and cross-disciplinary approaches in the multifaceted interplay between topology and singularity theory with an eye toward presenting the state of the art in an accessible and approachable way. Specific topics include braid groups and braid monodromy, combinatorics and topology of hyperplane arrangements, embedded curves and surfaces, moduli space theory, Hodge index theory, and the study of arithmetic, topological, and algebraic invariants. This book is published in cooperation with Real Sociedad Matematica Espanola.