  We derive a fully covariant theory of the mechanics of active surfaces. This
theory provides a framework for the study of active biological or chemical
processes at surfaces, such as the cell cortex, the mechanics of epithelial
tissues, or reconstituted active systems on surfaces. We introduce forces and
torques acting on a surface, and derive the associated force balance
conditions. We show that surfaces with in-plane rotational symmetry can have
broken up-down, chiral or planar-chiral symmetry. We discuss the rate of
entropy production in the surface and write linear constitutive relations that
satisfy the Onsager relations. We show that the bending modulus, the
spontaneous curvature and the surface tension of a passive surface are
renormalised by active terms. Finally, we identify novel active terms which are
not found in a passive theory and discuss examples of shape instabilities that
are related to active processes in the surface.
