Let K be a compact subset of the complex plane, C(K) be the set of all continuous functions on K and R(K) be the algebra of functions in C(K) which are uniform limits of rational functions without poles on K. For \(\phi\) smooth and \(f\in C(K)\), define \[ (T_{\phi}f)(w)=\phi (w)f(w)+(2\pi i)^{-1}\iint _{K}f(z)(z-w)^{-1}(\partial \phi /\partial \bar z)d\bar z\wedge dz. \] A closed subalgebra A of C(K) is said to be a (planar) T-invariant algebra if R(K)\(\subseteq A\) and \(T_{\phi}A\subseteq A\) for all smooth \(\phi\) with compact support. R(K) and A(K) \(\{=C(K)\cap Hol(K^ 0)\}\) are the prototypes of T-invariant algebras. Introducing on an arbitrary non-compact Riemann surface, a Cauchy transform operator which acts on measures and solves an inhomogeneous \({\bar \partial}\)-equation, we proceed to give an outline of a theory of T-invariant algebras on Riemann surfaces with appliations to meromorphic approximation theory. Information on Gleason parts is also obtained. Further developments will appear elsewhere.