This book offers an overview of selected topics on the topology of singularities, with emphasis on its relations to other branches of geometry and topology. The opening chapters cover complex singularities and collect a myriad of results spread in a vast literature, including recent research. The second part of the book studies real analytic singularities which arise from the topological and geometric study of holomorphic vector fields and foliations. In the low dimensional case these turn out to be related to fibred links in the 3-sphere defined by meromorphic functions.