  In the recent years, a lot of attention has been paid to the development of
solid foundations for the composition and inversion of schema mappings. In this
paper, we review the proposals for the semantics of these crucial operators.
For each of these proposals, we concentrate on the three following problems:
the definition of the semantics of the operator, the language needed to express
the operator, and the algorithmic issues associated to the problem of computing
the operator. It should be pointed out that we primarily consider the
formalization of schema mappings introduced in the work on data exchange. In
particular, when studying the problem of computing the composition and inverse
of a schema mapping, we will be mostly interested in computing these operators
for mappings specified by source-to-target tuple-generating dependencies.
