In this thesis we study the classi cation problem of boundary singularities of functions in symplectic and volume-preserving geometry. In particular we generalise several well known theorems concerning the classi cation of isolated singularities of functions and volume forms in the presence of a \boundary", i.e. a germ of a xed smooth hypersurface. The results depend in turn on a generalisation of the relative de Rham cohomology and the corresponding Gauss-Manin theory to the case of isolated boundary singularities and in particular, on a relative version of the so called Brieskorn-Deligne-Sebastiani theorem, concerning the niteness and freeness of certain cohomology modules.