  Let X be a smooth complex variety and Y be a closed subvariety of X, or more
generally, a closed subscheme of X. We are interested in invariants attached to
the singularities of the pair (X, Y). We discuss various methods to construct
such invariants, coming from the theory of multiplier ideals, D-modules, the
geometry of the space of arcs and characteristic p techniques. We present
several applications of these invariants to algebra, higher dimensional
birational geometry and to singularities.
