Yamabe problems for formally self-adjoint, conformally covariant, polydifferential operators

Formally self-adjoint, conformally covariant, polydifferential operators provide a general framework for studying variational problems, such as prescribing the scalar, $Q$-, or $\sigma_2$-curvatures, within a conformal class. We describe recent progress on Yamabe problems for such operators, including uniqueness results on the sphere and nonuniqueness results in general. We also highlight a number of open questions related to these operators, some of which constitute a possible blueprint for the general solution of the Yamabe problem for polydifferential operators.