In the first part, I will introduce vanishing cycles from a topological perspective, using the topology of complex hypersurface singularity germs as a motivation. The second part will be devoted to applications: (a) I plan to show how vanishing cycles can be used to compute the Euler characteristic of a complex projective hypersurface with arbitrary singularities; (b) I will briefly explain Beilinsonâs gluing of perverse sheaves via vanishing cycles; (c) I will indicate how vanishing cycles can be used to categorify Donaldson-Thomas invariants; (d) If time permits, I will explain how motivic characteristic classes for vanishing cycles can be used in birational geometry to detect jumping coefficients of multiplier ideal and to characterize rational or du Bois singularities.