Vanishing cycles appear naturally in the picture when studying families of hypersurfaces, usually regarded as singular fibrations. The behaviour of vanishing cycles seems to be the cornerstone for understanding the geometry and topology of such families of spaces. There is a large literature, mostly over the last 40 years, showing the various ways in which vanishing cycles appear. For instance, we may associate to a holomorphic function f its sheaf of vanishing cycles, encoding information about the singularities and the monodromy of f.

Although quite sophisticated information is available (e.g. in Hodge theoretic terms, see the survey [Di2]), there are many open questions on the geometry of vanishing cycles (see for instance Donaldson's paper [Don] for an intriguing one).

This book proposes a systematic geometro-topological approach to the vanishing cycles appearing especially in nonproper fibrations. In such fibrations, some of the vanishing cycles do not correspond to the singularities on the space. Nevertheless, if the fibration extends to a proper one, then new singularities appear at the boundary and their relation to the original context may explain the presence of those vanishing cycles. The study of this type of problem in the setting of singular spaces and stratified singularities started notably with the works of Goresky and MacPherson, Hamm and Lê.