Let $X$ be a compact Riemann surface of genus $g>1$. Is there any reference that completely classifies all holomorphic line bundles on $X$?
Moreover, for any line bundle $L$ on $X$, can we compute the dimension of the cohomology $H^i(X,L)$ for $i \ge 0$？
The notion of what it might mean to completely classify changes in this situation. In the case of $\mathbb{P}^1$, we know all the line bundles: they are of the form $\mathcal{O}(n)$ for $n\in \mathbb{Z}$ and moreover, they form a group isomorphic to $\mathbb{Z}$.
In the higher genus $(\ge 1)$ cases, there are uncountably many line bundles and the best way to keep track of them is via a moduli space, called $\mathrm{Pic}(X)$. $\mathrm{Pic}(X)$ for a curve fits into a short exact sequence $$ 0\to \mathrm{Pic}^0(X)\to \mathrm{Pic}(X)\to \mathbb{Z}\to 0 $$ where $\mathrm{Pic}^0(X)$ is the subgroup of degree $0$ line bundles. For $g(X) =1$, i.e. $X$ elliptic, $\mathrm{Pic}^0(X) \cong X$. This is in Hartshorne for instance. For $g(X) \ge 2$, the situation is more complicated. The $\mathrm{Pic}^0(X)$ all exist as projective varieties. In particular, they are all Abelian varieties of rank $g$ - or if you like compact complex Lie groups : hence complex tori of the form $\mathbb{C}^g/\Lambda$.
So, what it means to classify holomorphic line bundles on a curve $X$ with $g≥1$ has a different solution than many other classification problems.