  Classical W-gravities and the corresponding quantum theories are reviewed.
W-gravities are higher-spin gauge theories in two dimensions whose gauge
algebras are W-algebras. The geometrical structure of classical W-gravity is
investigated, leading to surprising connections with self-dual geometry. The
anomalies that arise in quantum W-gravity are discussed, with particular
attention to the new types of anomalies that arise for non-linearly realised
symmetries and to the relation between path-integral anomalies and non-closure
of the quantum current algebra. Models in which all anomalies are cancelled by
ghost contributions lead to new generalisations of string theories.
