Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well.

Suppose that we are given a monodromy representation $\rho: \pi_1(X,x) \longrightarrow \text{GL}(V)$.

How does one compute the cohomology $H^i(X,L)$ of the local system $L$ associated to $\rho$?