  The main result of this paper is the construction of a conformally covariant
operator in two dimensions acting on scalar fields and containing fourth order
derivatives. In this way it is possible to derive a class of Lagrangians
invariant under conformal transformations. They define conformal field theories
satisfying equations of the biharmonic type. Two kinds of these biharmonic
field theories are distinguished, characterized by the possibility or not of
the scalar fields to transform non-trivially under Weyl transformations. Both
cases are relevant for string theory and two dimensional gravity. The
biharmonic conformal field theories provide higher order corrections to the
equations of motion of the metric and give a possibility of adding new terms to
the Polyakov action.
