This semester I took a lecture on Riemann surfaces. The professor proved the Riemann-Roch theorem (stated below). As an application of it, he proved elementary results, we did earlier in the course anyway, e.g. that the Riemann-sphere has genus $0$.
My question is: What are the (less elementary) applications and implications of the Riemann-Roch theorem in the form below?
Theorem: Let $X$ be a compact Riemann-surface, $D$ a divisor of $X$. Then the Cech-cohomology groups $H^0 (X, \mathcal{O}_D)$ and $H^1(X, \mathcal{O}_D)$ are finite-dimensional vector-spaces and $$\dim H^0 (X, \mathcal{O}_D) - \dim H^1(X, \mathcal{O}_D) = 1 - g + \deg D$$ where $\mathcal{O}_D$ is the sheaf with $\mathcal{O}_D(U) := \{ f \text{ meromorphic on } X : ord_x(f) \geq -D(x), \forall x \in U \}$ and the natural restriction maps, and $g$ is the genus of $X$.