  This paper re-develops the Nevanlinna theory for meromorphic functions on
$\mathbb C$ in the viewpoint of holomorphic forms. According to our
observation, Nevanlinna's functions can be formulated by a holomorphic form.
Applying this thought to Riemann surfaces, one then extends the definition of
Nevanlinna's functions using a holomorphic form $\mathscr S$. With the new
settings, an analogue of Nevanlinna theory on \emph{weak $\mathscr S$-exhausted
Riemann surfaces} is obtained, which is viewed as a generalization of the
classical Nevanlinna theory on $\mathbb C$ and $\mathbb D.$
