Since no one has done it so far let me also mention the Riemann-Roch theorem for complex analytic or algebraic curves. (True, if one is mainly interested in the analytic case, then the algebraic version will not be of much use since to apply it one needs to show first that any Riemann surface is algebraic, which is most easily done by taking a projective embedding, which requires in turn the analytic Riemann-Roch theorem.) One needs only 1-variable complex analysis to define smooth compact complex curves and rational functions and divisors on them. Now if the degree of a divisor $D$ is $>2g-2$ where $g$ is the genus, then $\dim\mathcal{L}(D)=d-g+1$ where $d=deg(D)$. One does not even need the canonical divisor to state this.
As a consequence one gets many results on meromorphic functions on Riemann surfaces. For instance
there exist nonconstant meromorphic functions.
the Mittag-Leffler problem (find a meromorphic differential form with given poles and given principal parts at the poles) has a solution iff the sum of the residues is 0. There is a similar statement for meromorphic functions that can be proved in a similar way.
any algebraic curve (or Riemann surface) is projective and moreover can be embedded in $\mathbf{P}^3$.