  Constrained Willmore surfaces are conformal immersions of Riemann surfaces
that are critical points of the Willmore energy $W=\int H^2$ under compactly
supported infinitesimal conformal variations. Examples include all constant
mean curvature surfaces in space forms. In this paper we investigate more
generally the critical points of arbitrary geometric functionals on the space
of immersions under the constraint that the admissible variations
infinitesimally preserve the conformal structure. Besides constrained Willmore
surfaces we discuss in some detail examples of constrained minimal and volume
critical surfaces, the critical points of the area and enclosed volume
functional under the conformal constraint.
