  We derive a fully covariant theory of the hydrodynamics of nematic and polar
active surfaces, subjected to internal and external forces and torques. We
study the symmetries of polar and nematic surfaces and find that in addition to
5 different types of in-plane isotropic surfaces, polar and nematic surfaces
can be classified into 5 polar, 2 pseudopolar, 5 nematic and 2 pseudonematic
types of surfaces. We give examples of physical realisations of the different
types of surfaces we have identified. We obtain expressions for the equilibrium
tensions, moments, and external forces and torques acting on a passive polar or
nematic surface. We calculate the entropy production rate using the framework
of thermodynamics close to equilibrium and find constitutive equations for
polar and nematic active surfaces with different symmetries. We study the
instabilities of a confined flat planar-chiral polar active layer and of a
confined deformable polar active surface with broken up-down symmetry.
