  Motivated by the unique physical properties of {\em biological active
matter}, e.g., cytoskeletal dynamics in eukaryotic cells, we set up {\em
effective} two-dimensional (2d) coarse-grained hydrodynamic equations for the
dynamics of thin {\em active gels} with polar or nematic symmetries. We use the
well-known three-dimensional (3d) descriptions [K. Kruse {\em et al}, {\em Eur.
Phys. J E}, {\bf 16}, 5 (2005); A. Basu {\em et al}, {\em Eur. Phys. J E}, {\bf
27}, 149 (2008)] for thin active gel samples confined between parallel plates
with appropriate boundary conditions to derive the effective 2d constitutive
relations between appropriate thermodynamic fluxes and generalised forces for
small deviations from equilibrium. We consider three distinct cases,
characterised by spatial symmetries and boundary conditions, and show how such
considerations dictate the structure of the constitutive relations. We use
these to study the linear instabilities, calculate the correlation functions
and the diffusion constant of a small tagged particle, and elucidate their
dependences on the {\em activity} or nonequilibrium drive.
